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9
Transfer Matrix Method for Controlled Multibody Systems

9.1 Introduction

Research on the dynamics of multibody systems began more than 40 years ago and international commercial software based on a great many theories and methods for multibody systems has been developed. The transfer matrix method for multibody systems (MSTMM) does not need global dynamic equations of the system and has a lot of advantages, such as easy modeling, high programming stylization and high computational efficiency. In principle, the MSTMM can be combined with any other mechanics method, including all kinds of multibody system dynamics methods, the finite element method (FEM), the classic mechanics method and the analytical mechanics method, so that the advantages of different methods can be exploited. The MSTMM can be combined with all kinds of software to improve its computational speed. Precise performance prediction and effective controls of the multibody system are very important in control engineering problems, especially in fields such as the firing control of the launch system of high‐performance weapons, spatial structure control and robots. Because the problems are complex and the computation speed for the multi‐rigid‐flexible‐body system (MRFS) is too slow to satisfy the required response time for quick control, the dynamic models of multibody systems have to be simplified when dealing with those problems, and the model error needs to be compensated for through complex control design. The excessive simplification of the dynamic model usually results in difficulty when predicting the dynamic law of the system and the redundant high‐frequency behavior of the system, and therefore the control precision of the system is deeply affected. For example, when the firing control system of the launch system of a modern weapon is designed, the launch system is modeled as a simple multi‐rigid‐body system, where the deformation vibration of the system is not considered, resulting in poor control precision and low sensitivity. The best way to solve this problem is to introduce the dynamic method of a controlled multibody system which describes the dynamic of an MRFS perfectly and also satisfies the demand for rapid computation speed for real‐time control. The modeling of the multibody system and the controller are established synchronously and completely.

In this chapter, the following methods developed by the authors will be introduced: the mixed transfer matrix method for multibody systems and other multibody system dynamic methods [71, 72], the mixed MSTMM and the FEM [76], the finite segment discrete time MSTMM [81], the linear transfer matrix method (TMM) for controlled multibody systems [77], the discrete time TMM for controlled multibody systems [78] and the Riccati discrete time MSTMM [78, 80]. These methods make it easier to study the dynamics of multi‐dimensional systems and complex structures by using the MSTMM.

9.2 Mixed Transfer Matrix Method for Multibody Systems

In this section, the mixed TMM for multibody systems and other multibody system dynamic methods [71, 72] developed by the authors is introduced. In this method, any multibody system can be decomposed into subsystems, and the connection relation among the subsystems can be regarded as the boundary of each subsystem, as if the other subsystems do not exist. We usually establish the “global” dynamic equations using the ordinary method of multibody system dynamics for some subsystems and the “overall” transfer equation using the MSTMM for other subsystems. The unknown state variables on these “boundary subsystems” are considered to be external forces in the subsystem handled by the ordinary dynamic method and the internal forces in the subsystem handled by the MSTMM. Combining the global dynamic equations (usually hybrid differential‐algebra equations) with the overall transfer equations (algebra equations), the corresponding equations of the overall system can be assembled. Once all the global dynamic equations and the overall transfer equations of an MRFS are solved, the time history of the system dynamics can be obtained. It is an advantage of the method that each subsystem can adopt an optimum mathematic model and software.

The main steps for a numerical algorithm that can be used to solve MRFSs using the mixed MSTMM and other multibody system dynamics methods can be summarized as follows:

Give the system parameters and the initial state parameters etc.
Set the simulation step sequence number (discrete time step).
According to the angular coordinates of a rigid body and their first‐ and second‐order derivatives with respect to time at time , calculate the coefficients for the step‐by‐step time integration method at time t_i and estimate the angular coordinates of a rigid body at time t_i. Take this as the initial value of the next iterative procedure.
Set the iterative variable .
Regard the motion state of a rigid body as the boundary condition of a flexible body and compute the dynamic response of a flexible body using the multibody system dynamic method.
Set the iterative variable for the angular coordinates of the rigid body .
Compute the transfer matrices of each body element and hinge element.
Assemble the overall transfer equations according to the structure of the system.
Regard the boundary conditions of the system and the motion state and force state of the subsystem boundary as the boundary conditions of the overall transfer equations. Solve the overall transfer equations to obtain the state variable of each element at time t_i.
Judge the convergence criterion. If the square sum of the errors of all angular coordinates of rigid bodies from the kth to the ()th iterations is bigger than the required supposed precision value or the iterative loop does not reach the given iteration number, set and return to (7).
Compute the end‐point velocity, the acceleration of each element and the first‐ and second‐order derivatives with respect to time of the rigid‐body angular coordinates using the step‐by‐step time integration method.
Let . If j does not reach the required times, return to (5).
If the required simulation time has been reached, exit the program; if not, let and return to (3).

The following two examples are given to validate the proposed method.

Example 9.1

A multibody system is composed of one reticulated lumped mass‐spring system and four rigid bodies connected by smooth hinges, as shown in Figure 9.1. One end of the rigid body 1 is connected to the fixed point through the smooth hinge. The initial positions of this system are shown in Figure 9.1, and the initial velocities are zero. Determine the vertical‐plane motion of the system under the effect of gravity using the mixed MSTMM and other multibody system dynamics methods.

Solution

This system is decomposed into five subsystems: body subsystem 1, reticulated lumped mass‐spring subsystem 2, body subsystem 3, body subsystem 4 and body subsystem 5. The dynamic equations of reticulated lumped mass‐spring subsystem 2 are obtained by the Newton–Euler method, and for bodies 1, 3, 4 and 5 the corresponding transfer equations can be assembled by the TMM for multibody systems. As shown in Figure 9.2, the reticulated lumped mass‐spring subsystem 2 is a set of particles connected by springs. m_i,j denotes the lumped mass and its mass in the ith row and jth column, denotes the spring and its elastic coefficient between the jth column and the ()th column in the ith row, denotes the spring and its spring coefficient between the ith row and the ()th row in the jth column, and denotes the position vector of particle m_i,j at time t in the inertia coordinate system.

According to the force analysis of particles shown in Figure 9.3, the equations of motion of the set of lumped mass can be obtained

(a)

where , , and are the elastic forces of the springs, respectively, and is the external force.

The initial conditions are

(b)

where x_i,j,0 and y_i,j,0 are the initial position coordinates of the lumped mass m_i,j.

For , , and , the corresponding elastic forces of springs are zeros

(c)

As an example, is used to show the computation procedure of the elastic force of the spring

(d)

Denoting as the spring length at time t and as the initial length of the spring gives

(e)

(f)

(g)

According to Equation (g), the elastic forces of the other springs can be obtained as follows:

(h)

The external forces acting on the lumped mass m_1,1, m_1,5, m_5,1 and m_5,5 can be written as

(i)

The external force acting on the other lumped mass can be written as

(j)

The external forces acting on the “boundary” of the reticulated lumped mass‐spring subsystem 2 include the interaction forces between bodies 1, 3, 4 and 5 and subsystem 2. These external forces are part of the state variables of the boundary state of subsystems 1, 3, 4 and 5.

The transfer equations of bodies 1, 3, 4 and 5 can be written as

(k)

where , , and are the transfer matrices of bodies 1, 3, 4 and 5, respectively, and they are all the transfer matrices of a planar rigid body with one input and one output end, as shown in Equation (7.102). are the boundary state vectors of the system:

(l)

(m)

, , and are the boundary state vectors of the subsystems, and the state variables in these state vectors denote the internal forces that exist in the dynamic equations of the flexible body.

Substituting the boundary conditions into the transfer equations of the subsystem, the sum of the number of unknown variables in the independent algebra equations and the dynamic equations of the subsystem is equal to the number of equations. The system motion is determined by solving the simultaneous equations. The mass of each lumped mass is 0.04 kg and the elastic coefficient of each spring is 10 kN/m. Four rods are exactly the same, the mass center of each rod is its geometry center, the mass is 1 kg, the length is 1 m, each inertia moment with respect to the mass center is 1/12 kg · m², the initial velocity of each rod is zero, and the initial position is shown in Figure 9.1. The results obtained by using the mixed MSTMM and other multibody system dynamics methods are shown in Figure 9.4. Figure 9.4h shows that the conservation of mechanical energy holds, which indicates that the method used in this section has high computation stability and computation accuracy.

Example 9.3

An MRFS composed of rigid rods, springs and lumped mass is shown in Figure 9.8. One end of rigid body 2 is connected to fixed point through a smooth hinge. The system is horizontal and undeformed at the initial time, and the initial velocities are zero. Determine the vertical plane motion of the system under the effect of gravity using the mixed MSTMM and another multibody system dynamics method.

Solution

The MRFS is decomposed into three subsystems: the lumped mass‐spring subsystem 4, body subsystem 2 and body subsystem 6. The lumped mass‐spring subsystem 4 is computed by the Newton–Euler method, and bodies 2 and 6 can be computed by MSTMM.

Similarly to above example, the dynamic equations of lumped mass‐spring subsystem 4 are

(a)

where m_i is the mass of lumped mass i, is the elastic coefficient of the spring between the jth lumped mass and the ()th lumped mass, is the initial length of spring , and x_i(t) and y_i(t) are the position coordinates of lumped mass i in the inertia coordinate system at time t.

The initial conditions are

(b)

The transfer equations of subsystem bodies 2 and 6 are obtained by MSTMM

(c)

where and are the transfer matrices of subsystem bodies 2 and 6, respectively, as shown in Equation (7.101).

The boundary conditions of the system are

(d)

The boundary conditions of the subsystem are

(e)

The mass of each lumped mass is 0.2 kg, the elastic coefficient of each spring is 10 kN/m, the mass center of each rigid body is the geometry center, and the mass is 1 kg. The length is 1 m. Each inertia moment with respect to the mass center is 1/12 kg · m². The time history of the system is computed using the mixed method (the MSTMM and the Newton–Euler method) and the Newton–Euler method. The results are shown in Figure 9.9. In this figure, the numerical results obtained by the two methods show good agreement, which validates the efficiency of the mixed MSTMM and the Newton–Euler method for solving the dynamics of the MRFS.

Image described by caption and surrounding text. — **Figure 9.1** MRFS model with a reticulated particle–spring system.

Reticulated lumped mass–spring system with five rows and columns of circles labeled m1,1; m2,1; m3,1; m4,1; m5,1; m1,2; m2,2; m3,2; m4,2; m5,2; m5,3; m4,3; m3,3; etc. interconnected by springs (zigzag lines). — **Figure 9.2** Reticulated lumped mass–spring system.

Force analysis of a lumped mass, represented by a circle labeled ri,j with 4 springs labeled K(i–1,i),j; Ki,(j,j+1); K(i,i+1),j; and Ki,(j–1,j) and an outward arrow labeled mi,j g. — **Figure 9.3** Force analysis of a lumped mass.

**Figure 9.4** Dynamic simulation of a multibody system with a reticulated particle–spring system: (a) rotation angle of rigid body 1, (b) rotation angle of rigid body 5, (c) rotation angle of rigid body 3, (d) rotation angle of rigid body 4, (e) coordinates of particle 2, (f) coordinates of particle 17, (g) coordinates of particle 24 and (h) total mechanical energy of the system.

**Figure 9.5** Dynamic simulation of a multibody system with a plate: (a) rotation angle of rigid body 1, (b) rotation angle of rigid body 5, (c) rotation angle of rigid body 3, (d) rotation angle of rigid body 4, (e) coordinates of particle 2, (f) coordinates of particle 17, (g) coordinates of particle 24 and (h) total mechanical energy of the system.

Equivalent flexible plate with five rows and columns of circles interconnected by springs (zigzag lines). — **Figure 9.6** Equivalent flexible plate.

Graphs of ϑ2/rad and ϑ6/rad vs. t (top), coordinates of 2 and 3 vs. t (middle), and coordinates of 4 and 5 vs. t (bottom), each with 2 sets of 2 coinciding waves representing Newton–Euler method and the mixed method. — **Figure 9.9** Time history of the dynamics of an MRFS: (a) time history of the rotations of rigid body 2 and rigid body 6, (b) coordinates of particle 2 and particle 3, and (c) coordinates of particle 4 and particle 5.

9.3 Finite Element Transfer Matrix Method for Multibody Systems

The dynamics of linear multibody systems are solved by combining the MSTMM and the FEM, and the FEM is “reconstructed” by the MSTMM, or the MSTMM is “reconstructed” by the FEM. The mixed MSTMM and the FEM [76] introduced in this section is different from the “reconstructed method” mentioned above. In the proposed method, the two independent methods are applied in different subsystems of the same multibody system, that is, the multibody system is divided into subsystems and each of these subsystems could be readily modeled by the MSTMM or the FEM. The overall transfer equation is obtained by the MSTMM, and the global dynamic equations of other two‐ or three‐dimensional complex subsystems are obtained by the FEM. The subsequent processing method is the same as in section 9.2.

Example 9.4

As shown in Figure 9.10, the chain MRFS is composed of beam 1, smooth hinge 2, rigid rod 3, smooth hinge 4 and rigid rod 5. The left end of elastic rod 1 is fixed to the wall, and the right end of rod 5 is free. All the rods are initially placed in the horizontal position. Determine the dynamics of the system using the mixed MSTMM and the FEM.

Solution

The elastic rod 1 is regarded as subsystem 1 and computed by the FEM. Elements 2, 3, 4 and 5 are regarded as subsystem 2 and computed by the MSTMM.

Elastic rod 1 is regarded as an Euler–Bernoulli beam with only transverse deformation, the dynamic equation of beam 1 is established by the FEM, and the process is as follows.

As shown in Figure 9.11, the Euler–Bernoulli beam is divided into four elements, and the transverse displacement v(x, t) of any point on the beam can be denoted as

(a)

where ψ_i is the shape function and δ_i is the nodal coordinates.

Without loss of generality, δ₁ and δ₃ are regarded as the transverse deformations of nodes 1 and 2, respectively, , , thus ψ_i must satisfy the boundary conditions:

The strain energy and kinetic energy of a beam element are, respectively

(b)

(c)

where EI is bending stiffness, l is unit length and is the mass per unit length of the beam.

Let

According to the boundary conditions, we can obtain

Let

(d)

Substituting Equation (a) into Equations (b) and (c)

the element stiffness matrix and element mass matrix can be obtained

These equations can be expressed in matrix form as

(e)

The corresponding external force vector is

If we only consider the concentrated forces acting on the nodes, the node external force vector is

(f)

Therefore, the dynamic equation of beam 1 is

(g)

where

(h)

When solving Equation (h), the element node displacement matrix is needed. For the partition mode shown in Figure 9.11, we can obtain

In this matrix, the row number and column number correspond to the element node number and element number, respectively. The matrix element is the global node number, denoted by B. Let , and equal zero while computing, then the global stiffness matrix, global mass matrix and global external force vector can be obtained by the following loop:

DO I = 1,5
  k(2*B(1,i)-1:2*B(1,i),2*B(1,i)-1:2*B(1,i)) = k(2*B(1,i)-1:2*B(1,i),2*B(1,i)-1:2*B(1,i)) + kd(1:2,1:2)
  k(2*B(1,i)-1:2*B(1,i),2*B(2,i)-1:2*B(2,i)) = k(2*B(1,i)-1:2*B(1,i),2*B(2,i)-1:2*B(2,i)) + kd(1:2,3:4)
  k(2*B(2,I)-1:2*B(2,I),2*B(1,I)-1:2*B(1,I)) = k(2*B(2,I)-1:2*B(2,I),2*B(1,I)-1:2*B(1,I)) + kd(3:4,1:2)
  k(2*B(2,I)-1:2*B(2,I),2*B(2,I)-1:2*B(2,I)) = k(2*B(2,I)-1:2*B(2,I),2*B(2,I)-1:2*B(2,I)) + kd(3:4,3:4)
  m(2*B(1,i)-1:2*B(1,i),2*B(1,i)-1:2*B(1,i)) = m(2*B(1,i)-1:2*B(1,i),2*B(1,i)-1:2*B(1,i)) + md(1:2,1:2)
  m(2*B(1,i)-1:2*B(1,i),2*B(2,i)-1:2*B(2,i)) = m(2*B(1,i)-1:2*B(1,i),2*B(2,i)-1:2*B(2,i)) + md(1:2,3:4)
  m(2*B(2,I)-1:2*B(2,I),2*B(1,I)-1:2*B(1,I)) = m(2*B(2,I)-1:2*B(2,I),2*B(1,I)-1:2*B(1,I)) + md(3:4,1:2)
  m(2*B(2,I)-1:2*B(2,I),2*B(2,I)-1:2*B(2,I)) = m(2*B(2,I)-1:2*B(2,I),2*B(2,I)-1:2*B(2,I)) + md(3:4,3:4)
  p(2*b(1,I)-1:2*b(1,I)) = p(2*b(1,I)-1:2*b(1,I)) + pd(1:2)
  p(2*b(2,I)-1:2*b(2,I)) = p(2*b(2,I)-1:2*b(2,I)) + pd(3:4)
ENDDO

Discretize Equation (g) using the Newmark‐β method, thus

(i)

For subsystem 2, the transfer equations of elements 3, 4 and 5 can be developed by the MSDTTMM

(j)

where and are the transfer matrices of elements 3 and 5, respectively, which are rigid bodies with one input and one output end moving in a plane, as shown in Equation (7.101). is the transfer matrix of smooth hinge 4 moving in a plane which connects with the output ends of the rigid bodies and its internal torque is zero, as shown in Equation (7.230).

The overall transfer equation of the subsystem is

(k)

The overall transfer matrix is

(l)

The smooth hinge 2 is the boundary of subsystem 2, that is

(m)

The boundary conditions of the system are

(n)

Combining Equation (i) with Equation (j), we can obtain 14 equations for subsystems 1 and 2 with 14 unknown variables in them: , θ_3,2, q_x,3,2, q_y,3,2, x_6,5, y_6,5 and θ_6,5. Applying the external force and the initial conditions, the dynamic equations of the system can be solved.

The length of the homogeneous and elastic steel rod 1 is 1 m, and its cross‐section is square with length 1 cm. The density of rod 1 is and the Young’s modulus is . The configuration parameters of rigid rods 3 and 5 are the same, the mass is 1 kg, the length is 1 m, the mass center is the geometry center of the rod and the inertia moment with respect to the mass center is . The initial angles of the two rigid bodies are both π/3rad with respect to the horizontal line and the angular velocities are zero. The beam lies in the horizontal under the effect of gravity and its velocity is zero. The system dynamics are solved by the mixed method (MSTMM and FEM) and the Newton–Euler method, as shown in Figures 9.12–9.18. The transverse displacement of the connected point between rigid body 3 and the beam is given in Figure 9.12. The transverse displacements of nodes 1, 2 and 3 of the beam are given in Figures 9.13–9.15, respectively. The time histories of the angular motion of rigid bodies 3 and 5 are given in Figures 9.16 and 9.17, respectively. The time history of the interaction forces between rigid body 3 and beam 1 is illustrated in Figure 9.18.

Observation reveals that the results obtained by the mixed method (MTMM and FEM) and the Newton–Euler method are in good agreement, which validates the efficiency of the mixed MSTMM and the FEM for solving the dynamic response of the MRFS.

A horizontal bar divided into 5 segments with labels 0–5, with boxes labeled 1, 2, 3, and 4 below the second, third, fourth, and fifth bars from the left. Below the boxes is another bar with a 2-headed arrow labeled l. — **Figure 9.11** The element partition of beam 1.

Graph of y/m vs. t/s displaying 2 coinciding waves representing the mixed method (2 horizontal lines) and Newton–Euler method (“x” marker), illustrating the transverse displacement of the connected point. — **Figure 9.12** The transverse displacement of the connected point.

Graph of y/m vs. t/s illustrating the node 2 of the beam depicting transverse dashed waves representing the mixed method and "X" marks for Newton–Euler method. — **Figure 9.14** The transverse displacements of node 2 of the beam.

Graph of y/m vs. t/s illustrating the node 3 of the beam depicting transverse dashed waves representing the mixed method and "X" marks for Newton–Euler method. — **Figure 9.15** The transverse displacements of node 3 of the beam.

Graph of θ3/rad vs. t/s of the azimuth angle of rigid body 3 depicting transverse dashed waves representing the mixed method and "X" marks for Newton–Euler method. — **Figure 9.16** Time history of the azimuth angle of rigid body 3.

Graph of θ5/rad vs. t/s of the azimuth angle of rigid body 5 depicting transverse dashed waves representing the mixed method and "X" marks for Newton–Euler method. — **Figure 9.17** Time history of the azimuth angle of rigid body 5.

Graph of qy/N vs. t/s of the interaction forces between rigid body 3 and beam 1 depicting transverse dashed waves representing the mixed method and "X" marks for Newton–Euler method. — **Figure 9.18** Time history of the interaction forces between rigid body 3 and beam 1.

9.4 Finite Segment Transfer Matrix Method for Multibody Systems

This section introduces the finite segment discrete time transfer matrix method for multibody systems [81], which was developed by the authors. The beam is divided into finite rigid segments by the finite segment method, and the segments are connected by a torsional spring and a linear spring with a parallel damper. The inertia characteristic of the beam is described by a rigid segment, and the elasticity and damp characteristics are described by springs and dampers between the rigid segments, as shown in Figure 9.19. After being discretized by the finite segment method, the beam can be regarded as a chain‐type MRS connected by springs and dampers. The finite segment model is similar to the continuous beam model, with an increase in the number of rigid segments.

Finite segment model with a thick downward arrow labeled Equivalent from a beam to boxes connected by linear spring and damper. — **Figure 9.19** The finite segment model of a beam.

The stiffness coefficient of each spring connecting the segments can be determined as follows. The relation between the internal moment of the beam and the rotation angle generated by bending the beam can be obtained according to Euler–Bernoulli beam theory,

(9.1)

where M is the internal moment of the beam, EI is the bending stiffness and θ is the rotation angle generated by bending the beam.

Substituting difference for differential, we obtain

(9.2)

where the subscript i denotes the ith finite segment, Δ denotes difference, is the torsional stiffness coefficient of a torsional spring with respect to the corresponding finite segment and l_i is the length of the finite segment.

The elasticity of the finite segment is regarded as two series torsional springs. For a homogenous beam with uniform cross‐section, it becomes

(9.3)

where and are the stiffness coefficients of two series torsional springs corresponding to .

According to the finite segment model, the torsional springs and are introduced for two rigid segments of the series. The torsional stiffness coefficient between two rigid segments is

(9.4)

The moment and rotation angle between the two rigid segments can be obtained from

(9.5)

where is the elastic moment between two rigid segments, and and θ_i are the orientation angles of the ()th rigid segment and the ith rigid segment, respectively.

Similarly, considering the torsion and compression deformation of the beam, we obtain

(9.6)

where is the elastic force between rigid segments, namely the internal force between rigid segments, and and x_i are the position coordinates of the hinge point of the ()th rigid segment and the ith rigid segment, respectively.

Similar to the above derivation, the elastic coefficient of the linear spring between two rigid segments is

(9.7)

where EA is the extensional rigidity coefficient.

Example 9.5

A plane‐motion beam is mounted on a rotary base, as shown in Figure 9.20. The length of the beam is , the cross‐section area is and the Young’s module is . The inertia moment of the cross‐section area is and the density is . The angular velocity of the rotary base is

(9.8)

where Ω is the steady‐state angular velocity, and T is the accelerative time . Solve the equations of motion and determine the driving force for the motion.

Solution

The flexible beam is divided into N rigid segments connected by the elastic hinge and damper hinge. According to the boundary conditions

(9.9)

(9.10)

and the overall transfer equation, the linear equation system about the unknown variables in state vector of the base motion, can be obtained

(9.11)

where θ is the rotation angle, obtained by integration of Equation (9.8), and u_i,j is the element in the overall transfer matrix.

The vector can be obtained by solving Equation (9.11), where m_0,1 is the required driving moment. The results of the system dynamics obtained by finite segment MSDTTMM are shown in Figures 9.21–9.23. The time history of driving moment which the system needs to realize the expected motion is shown in Figure 9.21. The time history of the transverse deformation and longitudinal deformation of the tip end of the beam are shown in Figures 9.22 and 9.23, respectively. The solid line denotes the computation results by the Kane method [250] and ⋄ denotes the computation results by finite segment MSDTTMM. The results obtained by the two methods have good agreement. The computational speed of the finite segment MSDTTMM is much faster and it has higher computational efficiency.

For the converse problem, if the driving moment as shown in Figure 9.21 is known, we can determine the motion rule of the system. In this case, the state variable θ in the boundary condition (Equation (9.9)) will be treated as an unknown quantity. However, the state variable m may be considered as a known quantity. Solving the overall transfer equation, the motion rule can be obtained. The computation results are shown in Figure 9.24. The solid line denotes the time history of the expected angular velocity, and ⋄ represents the time history of angular velocity computed according to the driving moment, as shown in Figure 9.21.

The relation between computation time and numbers of segments (degrees of freedom) by finite segment MSDTTMM is shown in Figure 9.25. The computation time is proportional to the number of segments, while the computation time of the ordinary dynamic methods and the number of system degrees of freedom have an exponential relationship. This is the reason why the computing speed is faster by finite segment MSDTTMM than by other methods.

Graph of drive moment/N.m vs. t/s of the driving moment, displaying a bell-shaped curve representing Kane method with diamond markers for finite segment MSDTTMM. — **Figure 9.21** Time history of the driving moment.

Graph of deformation/(m) vs. t/s of a tip end, displaying an inverted bell-shaped curve representing the Kane method with diamond markers for finite segment MSDTTMM. — **Figure 9.22** Time history of the transverse deformation of a tip end.

Graph of the longitudinal deformation of a tip end displaying an inverted bell-shaped curve from the origin representing the Kane method with diamond markers for finite segment MSDTTMM. — **Figure 9.23** Time history of the longitudinal deformation of a tip end.

Graph of time history of rotation angular velocity displaying an ascending curve representing Kane method with diamond markers for finite segment MSDTTMM. — **Figure 9.24** Time history of rotation angular velocity.

Graph of the relation between computational time and number of segments with an ascending line from 0.16 of the origin with square markers. — **Figure 9.25** Relation between computational time and number of segments.

9.5 Transfer Matrix Method for Controlled Multibody Systems I

In reference [251], the control signal is regarded as the state variable of a state vector. The TMM is extended to control of a chain multibody system for modal analysis of the controlled multi‐link mechanical arms. The computational method for the dynamics of a real‐time controlled system, whose control force is related to the current system state, is introduced in this section. By taking the control characteristic parameters of the system as special mechanical parameters and re‐deriving the transfer matrices of controlled elements, there is no need to add the state variables corresponding to the control signal. The dynamics of controlled multibody systems can be studied by the MSTMM [77]. For a time‐delay controlled multibody system, the control force can be denoted by the previous system state and regarded as an external force, and only the control forces in the terms of the external force column matrix of the MSTMM need to be considered. The dynamics of the controlled system can be solved using the same methods of the MSDTTMM.

9.5.1 Transfer Matrix Method for Linear Real‐time Controlled Multibody Systems

9.5.1.1 Dynamic Model of Linear Controlled Multibody Systems

For the linear controlled multibody system shown in Figure 9.26, K_j and denote the elastic coefficient of the spring and viscous damping coefficient of element j, respectively. The simple harmonic external force with frequency Ω acting on lumped mass m_j is

(9.12)

Dynamic model of a linear controlled multibody system with series of boxes labeled m2, m4, mp, mk, and m2n connected by spring and damper, with rightward arrows at the bottom for x2,1; x4,3; xp,p−1; xk,k−1; etc. — **Figure 9.26** Dynamic model of a linear controlled multibody system.

The sensor of the controlled system is fixed on lumped mass m_k and the real‐time control force acting on the lumped mass m_p produced by the controller is

(9.13)

9.5.1.2 Transfer Equation and Extended Transfer Matrix of Elements

According to the method given in Section 2.7, when the system is in steady‐state motion, the motion of every lumped mass may be indicated by

(9.14)

where is the complex amplitude of the steady‐state forced vibration.

The complex amplitude of the simple‐harmonic external force can be written as

(9.15)

Substituting Equation (9.14) into Equation (9.13), the control force of the controlled element is

(9.16)

The complex amplitude of the control force is

(9.17)

According to the geometrical relationship and the force balance condition of the two ends of the controlled lumped mass p, we can obtain

(9.18)

Substituting Equation (9.17) into Equation (9.18), these equations can be written in the following matrix form

(9.19)

Thus, the transfer equation of the real‐time controlled element p is

(9.20)

The extended transfer equation of other elements without control is

(9.21)

where

(9.22)

9.5.1.3 Overall Transfer Equation and Overall Transfer Matrix

From Equations (9.20) and (9.21), we obtain

(9.23)

The second formula of Equation (9.23) yields

(9.24)

Substituting Equation (9.24) into the first formula of Equation (9.23), the overall transfer equation is obtained

(9.25)

where the overall transfer matrix is

(9.26)

9.5.1.4 Solving the Steady‐state Motion of the System

The boundary conditions of the system are

(9.27)

Substituting Equation (9.27) into Equation (9.25) yields

(9.28)

Solving Equation (9.28), the unknown state variables in the state vector of the system boundary can be obtained

(9.29)

By using Equation (9.21) in sequence and Equation (9.20), the state vectors at any point of the system can be determined. Therefore, the complex amplitude of each lumped mass can be obtained. From Equation (9.14), the steady‐state motion can be obtained

(9.30)

Example 9.6

For the linear controlled multibody system shown in Figure 9.27, let

The control signal comes from lumped mass 6. The control force acts on lumped mass 2, namely with and . Solve the steady‐state response of the system using the multibody system dynamics method and the TMM for linear controlled multibody systems.

Solution

Multibody system dynamics method
The overall dynamic equation of the system is

(a)

where

Because the values of initial conditions will not affect the steady‐state response of the system, they can be supposed to be zero

(b)
TMM for linear controlled multibody systems
Substituting parameters into Equations (9.19) to (9.29), by matrix manipulation the steady‐state motion of the system can be determined to be

(c)

To illustrate the influence of the real‐time control force on the motion of the system, the dynamics of the system with control () and without control () are computed by the TMM for linear controlled multibody systems and the multibody system dynamics method, respectively. The computation results are shown in Figure 9.28. The time history of the coordinate with control by the TMM for linear controlled multibody systems and the multibody system dynamics method are denoted by ◆ and —, respectively. The time history of the coordinates without control by the multibody system dynamics method is denote by × .

The results obtained by the two methods are in good agreement, which validates the efficiency and accuracy of the TMM for a linear controlled multibody system for solving the dynamics of controlled multibody systems. After the implementation of control, the amplitude of the steady‐state motion of the system is less than in the system without control.

It is convenient to obtain the frequency response of the linear controlled multibody system shown in Figure 9.27 using the TMM for a linear controlled multibody system, as shown in Figure 9.29. The amplitude and frequency characteristics of the system are changed because the controlled loop is considered, namely the vibration characteristics of the system are changed.

Example 9.7

The linear controlled multibody system in Figure 9.30 is similar to that in Example 9.4. The control force acts on a spring hinge and damper hinge 3, and the parameters of the controller are different; all other parameters are the same as those in Example 9.4. The control signal comes from lumped mass 6, and the control force acts on the spring hinge and damper hinge 3, namely with , and . Solve the steady‐state response of the system using the multibody system dynamics method and the TMM for a linear controlled multibody system.

Solution

Multibody system dynamics method
The overall dynamic equation of the system is

(a)

where

Since the initial conditions will not affect the steady‐state response of the system, we select

(b)
Transfer matrix method
The transfer equation of controlled element 3 is

(c)

where

The transfer equations of the elements without control are

(d)

Then

(e)

where the overall transfer equation is

(f)

The overall transfer matrix is

(g)

Applying the boundary conditions and the given parameters, the steady‐state response of the system is determined as

(h)

The steady‐state response of the system is computed using the TMM for a linear controlled multibody system and the multibody system dynamics method, as shown in Figure 9.31. The results for the steady‐state motion of the system obtained by the two methods are in good agreement, which validates the TMM of linear controlled multibody systems for solving the steady‐state motion of controlled multibody systems.

Example 9.8

A controlled branch multibody system is shown in Figure 9.32. Elements 1, 2, 3 and 4 are spatial springs, element 5 is a rigid body, elements 6, 7 and 8 are torsional springs, and elements 9, 10 and 11 are rotors. Element 5 is a feedback element whose motion is controlled by its state variable feedback. The stiffnesses of springs 1, 2, 3 and 4 are

The mass of element 5 is 2 kg, and the inertia moment with respect to the geometric center is

The mass of element 9 is 0.2 kg and the inertia moments of elements 9, 10 and 11 are all . The torsional stiffnesses of elements 6, 7 and 8 are , and , respectively. The coefficients of the proportional damping system are α =0.025 and β =0.05. The external moment of elements 9, 10 and 11 are

The angular displacement and angular acceleration are the feedback parameters, and the form of the control is

Solution

The state vectors of the connection point are defined as

(9.31)

(9.32)

(9.33)

, , , , , , and have the same forms, , and have the same forms, and , , , , and have the same forms:

(9.34)

(9.35)

The overall transfer equation of the system is

(9.36)

where

(9.37)

is composed of the state vectors at the boundary points of the system and is the overall transfer matrix of the system with 24 × 54 dimension.

The detailed expressions of the components of the controlled element are

The control parameters are

The boundary conditions of the system are

Eliminating the zero elements in the state vector yields the state vector , and deleting the rows in corresponding to the zero elements in leads to the 24 order square matrix :

(9.38)

is related to the structure parameters of the system and the natural frequency ω_k. When the structure parameters of the system are determined, the value of the determinant of matrix corresponding to the eigenfrequency ω_k is zero, that is

(9.39)

Equation (9.39) is the eigenequation of the system. Solving the eigenequation, the eigenfrequency of the system ω_k can be obtained. Under the conditions of given normalization, solving corresponding to the eigenfrequency ω_k, the state vectors , , , and can be obtained. Then according to the body dynamic equations of elements, the dynamic response of the system is determined.

The responses of this system with or without control are computed by the TMM for linear controlled multibody systems and the Newton–Euler method, respectively. The results are shown in Figures 9.33–9.35. From the figures it can be seen that the results obtained by two methods are in good agreement. The steady‐state vibration amplitude of the controlled system is smaller than that of the non‐controlled system, which validates the capability of the TMM for linear controlled multibody systems to solve the dynamics problem of a controlled branch multibody system.

Linear multibody vibration system with 3 boxes labeled m2, m4, and m6 connected with springs and dampers, with a controlled loop depicted by a line from box m6 to box m2. — **Figure 9.27** A linear multibody vibration system with a controlled loop.

Graphs of X21/m (top), X43/m (middle), and X65/m (bottom) vs. t/s, each has curves for multibody system dynamic method with diamond markers for transfer matrix method and “X” marks for non-controlled system. — **Figure 9.28** The computational results of system dynamics. (a) Time history of the location coordinate of lumped mass 2. (b) Time history of the location coordinate of lumped mass 4. (c) Time history of the location coordinate of lumped mass 6.

2 Graphs illustrating the frequency response for Ka = 0 (top) and Ka = 5.0 (bottom), each with 3 intersecting curves representing X2,1 (solid); X4,3 (dashed); and X6,5 (dash-dotted). — **Figure 9.29** Frequency response for a linear controlled multibody system with a controlled loop. (a) Frequency response for K_a = 0. (b) Frequency response for K_a = 5.0.

Linear controlled multibody system with 3 boxes for m2, m4, and m6 connected by strings and dampers labeled K1,C1; K3,C3; and K5,C5, with a controlled loop depicted by a line from m6 to K3,C3. — **Figure 9.30** Linear controlled multibody system with a controlled loop.

Graphs of X21/m (top), X43/m (middle), and X65/m (bottom) vs. t/s, each has dashed curves representing TMM for linear controlled multibody system and scattered “X” marks for multibody system dynamics method. — **Figure 9.31** Time history of the dynamics of a system: (a) time history of the displacement of lumped mass 2, (b) time history of the displacement of lumped mass 4 and (c) time history of the displacement of lumped mass 6.

Graph of time history of angular velocity of element 5 around x axis displaying a transverse wave and intersecting curve from the origin for Newton–Euler method, with diamond markers for TMM and “X” marks for MSTMM. — **Figure 9.33** Time history of the angular velocity of element 5 around the x axis.

Graph of time history of angular velocity of element 5 around y axis displaying a transverse wave and intersecting curve from the origin for Newton–Euler method, with diamond markers for TMM and “X” marks for MSTMM. — **Figure 9.34** Time history of the angular velocity of element 5 around the y axis.

Graph of time history of angular velocity of element 5 around z axis displaying a transverse wave and intersecting curve from the origin for Newton–Euler method, with diamond markers for TMM and “X” marks for MSTMM. — **Figure 9.35** Time history of the angular velocity of element 5 around the z axis.

9.5.2 Transfer Matrix Method for Linear Delay Controlled Multibody Systems

A linear controlled multibody system is shown in Figure 9.26. If the system has a time delay τ and steady‐state motion, the motion of a lumped mass can be denoted by

(9.40)

where is the complex amplitude of the steady‐state forced vibration.

The control force is

(9.41)

Equation (9.41) can be written as

(9.42)

The complex amplitude of the control force is

(9.43)

Taking the known control force F_p,c of the current time into the transfer matrix of the controlled element, the transfer equation of the controlled element is

(9.44)

namely

(9.45)

The overall transfer matrix is

(9.46)

According to the boundary conditions of the system and the transfer relation, the complex amplitude of each lumped mass can be obtained. The steady‐state motion can be obtained from Equation (9.40):

(9.47)

3 Graphs illustrating X21/m (top), X43/m(middle), and X65/m (bottom) vs. t/s, each with transverse dashed curves representing TMM for linear multibody system and scattered “X” marks for Newton–Euler method. — **Figure 9.36** The time history of the motion of a controlled system with time delay: (a) time history of the displacement of lumped mass 2, (b) time history of the displacement of lumped mass 4 and (c) time history of the displacement of lumped mass 6.

9.6 Transfer Matrix Method for Controlled Multibody Systems II

In this section, using the system in Figure 9.37 as an example, the discrete time transfer matrix method (DTTMM) for controlled multibody systems [78], developed by the authors, is introduced.

Controlled multibody system with 5 boxes labeled m2, m4, mp, mk, and m2n connected by springs and dampers with line from mk to mp and 5 rightward arrows for x2,1; x4,3; xp,p–1; xk,k–1; etc. at the bottom. — **Figure 9.37** Controlled multibody system.

9.6.1 Discrete Time Transfer Matrix Method of Real‐time Controlled Multibody Systems

The DTTMM for multi‐rigid‐body systems was introduced in Chapter 7. The transfer equations of a lumped mass and a spring with longitudinal motion (see Figure 9.38) are obtained as follows

(9.48)

(9.49)

Image described by caption. — **Figure 9.38** The subsystem composed of a spring and a lumped mass.

The state vector is

(9.50)

The transfer matrix of the spring is

(9.51)

The transfer matrix of the lumped mass is

(9.52)

Linearizing the real‐time control force as shown in Equation (9.13), this can be written as

(9.53)

From the geometry relation and the force equilibrium condition of element p, we have

(9.54)

Substituting Equation (9.53) into Equation (9.54), the transfer equation of the controlled element p is

(9.55)

where

(9.56)

From the system structure we can obtain

(9.57)

From the second formula of Equation (9.57), we have

(9.58)

Substituting Equation (9.58) into Equation (9.57), the overall transfer equation is

(9.59)

The overall transfer matrix is

(9.60)

Substituting the boundary conditions and into Equation (9.53) yields

(9.61)

Hence

(9.62)

The state vector of each element in the system at any time can be solved using Equations (9.48) and (9.49) repeatedly, and Equation (9.55).

Example 9.12

The controlled planar flexible mechanism manipulator system is composed of hub 1, rectangular cross‐section beam 3 and fixed hinge 2 connected to elements 1 and 3, as shown in Figure 9.42a. The mass, radius of gyration and moment of inertia of hub 1 are , , , respectively. Hub 1 is driven by an electric motor and the expected time history of orientation angle is

(9.63)

To detect and control the vibration of the rectangular cross‐section beam, three pieces of piezoelectric ceramic (PZT) actuators/sensors are fixed onto its surface. The width and thickness of each PZT actuators/sensor are, respectively, the same as and much smaller than that of the beam. The positions of the PZT actuators/sensors on the corresponding flexible are (0 , 0.1), (0.45 , 0.55) and (0.9 , 1) in body‐fixed reference coordinates. The structure parameters of flexible arm 3 are , and . The parameters of the PZT actuators/sensors are all the same, namely , , and . The PZT actuators/sensors are placed on flexible arm 3 as shown in Figure 9.42b.

3 Graphs illustrating X21/m (top), X43/m (middle), and X65/m (bottom) vs. t/sof the system motion with or without control, each with transverse solid curves representing ka = 5.0 and scattered “X” marks for ka = 0.0. — **Figure 9.39** Time history of system motion with or without control: (a) time history of the displacement of lumped mass 2, (b) time history of the displacement of lumped mass 4 and (c) time history of the displacement of lumped mass 6.

Graphs of x21/m, x43/m, and x65/m vs. t/s of the system motion, each with transverse curves for Newton–Euler method, with diamond markers for TMM linear controlled and “X” marks for DTTMM controlled. — **Figure 9.40** Time history of system motion: (a) time history of the displacement of lumped mass 2, (b) time history of the displacement of lumped mass 4 and (c) time history of the displacement of lumped mass 6.

The following assumptions are made for system modeling:

The thickness of the bond for each PZT actuator and the flexible arm is too thin to have a deep effect on dynamics of the system, thus is neglected.
The PZT actuators are rigidly bonded to the flexible arm.
The thickness of each PZT actuator is transparent, and their effects on the mass distribution and stiffness distribution of system are neglected. The flexible arm is considered to be a uniform cross‐section Euler–Bernoulli beam.
The voltage of each PZT actuator is uniformly distributed. The polarization direction of each PZT is the same as the transverse vibration direction of the corresponding flexible arm. The trajectory tracking and active vibration control of the flexible arm are studied by combining the proportional‐differential (PD) feedback control and the modal velocity feedback control.

The control moments acting on the hub rigid body and the driving voltage applied to the ith segmented PZT are, respectively

(9.64)

(9.65)

where and are the desired angle and angular velocity of hub rigid body 1, respectively, θ₁ and are the actual angle and angular velocity, respectively, K_p and K_v are the proportion gain and velocity gain, respectively, τ₀(t) is the control moment for drive motor, K_ai is the gain coefficient of the ith segmented PZT actuator, V_i is the driving voltage applied to the ith segmented PZT actuator, u is the transverse deformation of the flexible arm, Y^k(x₂) is the kth model function of the flexible arm, and . is the position coordinate of each PZT on the corresponding flexible arm in the body‐fixed reference coordinate system.

Considering only the control moment on the rigid body related to its feedback state, the transfer matrix of the controlled rigid body vibrating in a plane is

(9.66)

where

u₄₁, u₄₂, u₄₅, u₄₆, u₅₇ and u₆₇ are the same as in Equation (7.101).

Using Equations (9.66) to (9.68), and considering the control moment of the PZT actuators, the transfer matrix of the controlled piezoelectric beam vibrating in a plane is

(9.67)

where

b and t_b are the width and thickness of the flexible arm, respectively. t_a is the thickness of the segmented PZT actuator.

For the fixed hinge whose inboard body is a rigid body and whose outboard body is a piezoelectric beam with large planar motion the position coordinates, orientation angles, internal moments and internal forces of the rigid body’s input and output are equal. The transfer relation between the generalized coordinates describing the deformation of the outboard flexible body and the input end of the fixed hinge only needs to be determined. The dynamics equation of the transverse vibration of the beam is given by Equation (8.91) and the highest order of the mode shapes is chosen to be 3. The distributed moments of the PZT actuators and the control equation of the PZT actuators should be taken into consideration when we deduce the transfer equation of the beam. The state vectors of the fixed hinge whose input end is a rigid body and output end is a piezoelectric beam with large planar motion are defined as

(9.68)

The transfer equation is

(9.69)

The transfer matrix is

(9.70)

where

For compound control by the PD controller and modal velocity feedback control based on PZT actuators, the transfer matrix of the control element is

(9.71)

where is the feedback parameter matrix related to the feedback parameter from feedback beam 3 to hub 1.

For the modal velocity feedback control, we have

(9.72)

According to the DTTMM for controlled multibody systems, the state vectors of the system are

(9.73)

The boundary conditions are

(9.74)

The overall transfer equation is

(9.75)

where , , and are determined by Equations (9.67), (9.70), (9.66) and (9.71), respectively.

The dynamics of the controlled flexible manipulator system calculated by the DTTMM for controlled multibody systems are shown in Figure 9.43. The time history of the expected orientation angle of the body‐fixed reference system of flexible arm 3 is shown in Figure 9.43a. The time history of the transverse deformation at the tip of arm 3 with and without PZT active control is shown in Figure 9.43b. The driving voltage applied to segmented PZT 1 is shown in Figure 9.43c. The simulation results show that the orientation angle of the flexible manipulator and expected track have good agreement. The transverse deformation at the tip of the arm decreases quickly. This validates the DTTMM for controlled multibody systems for the solution of the trajectory tracking of the flexible manipulator and active control of vibration.

Example 9.13

The system shown in Figure 9.44 is composed of a rigid body with one input and one output end and a spatial elastic hinge, and the effect of damping is not considered. Solve the dynamic response of the system under an external force acting on the rigid body.

The state vector of the input end of element 1 is defined as

(9.76)

The state vector of the output end of feedback element 2 is defined as

(9.77)

The transfer matrix of the control element is

(9.78)

where ,

The control parameters are

The transfer relation of control is

(9.79)

The overall transfer matrix is

(9.80)

where is the transfer matrix of the spatial elastic hinge and is the transfer matrix of the rigid body moving in space. is the coordinate transformation matrix around the z axis.

The boundary conditions are , , , , , , , , , , and .

The dynamics of the controlled multibody system are simulated by the DTTMM for controlled multibody systems and the Newton–Euler method. The simulation results are shown in Figure 9.45. From Figure 9.45, it can be seen that the system vibrates with equal amplitude under an external force without control, and the system motion attenuates to the equilibrium balancing position under feedback control. The results obtained by the two methods are in good agreement, which validates the DTTMM for controlled multibody systems for solving the dynamics of controlled multibody systems.

**Figure 9.43** Dynamics of a controlled flexible manipulator system: (a) time history of the orientation angle of a flexible arm, (b) time history of the transverse deformation at the tip of the arm and (c) time history of the driven voltage applied to segmented PZT 1.

Multibody system with a cuboid labeled 2 with an arrow to a box for K, leading to a circle with 2 ellipses intersecting inside labeled 1, linked to the ground symbol for 0. — **Figure 9.44** Multibody system composing of a rigid body and an elastic hinge moving in space.

Graph θz/rad vs. t/s of rigid body around z axis displaying a transverse wave for Newton–Euler method with intersecting curve from the origin, with diamond markers for TMM controlled and “X” marks for MSTMM. — **Figure 9.45** Time history of the angle of a rigid body around the z axis.

9.6.2 Discrete Time Transfer Matrix Method for Controlled Multibody Systems with Time Delay

For a controlled system with time delay, the control force in Equation (9.13) with time delay can be described as the function of the former motion parameters , and as follows

(9.81)

where τ is the time delay.

For the controlled system with time delay, substituting Equation (9.81) into Equation (9.54) and replacing u₂₃ of the control transfer matrix by the known control force F_p,c in current time yields

For the controlled system with time delay, the current control force is the function of the former force and can be regarded as an external force. The elements of are all known functions of the previous time step, and Equation (9.60) can be written as

(9.83)

Applying the boundary conditions of the system and using Equations (9.48) and (9.49) repeatedly, the state vectors of each element at any time can be obtained.

Graphs illustrating x21/m, x43/m, and x65/m vs. t/s of the system motion, each with transverse curves for Newton–Euler method with diamond markers for TMM linear controlled and “X” marks for DTTMM controlled. — **Figure 9.46** Time history of system motion: (a) time history of the displacement of lumped mass 2, (b) time history of the displacement of lumped mass 4 and (c) time history of the displacement of lumped mass 6.

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Table of Contents for 9 Transfer Matrix Method for Controlled Multibody Systems

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9.1 Introduction

9.2 Mixed Transfer Matrix Method for Multibody Systems

9.3 Finite Element Transfer Matrix Method for Multibody Systems

9.4 Finite Segment Transfer Matrix Method for Multibody Systems

9.5 Transfer Matrix Method for Controlled Multibody Systems I

9.5.1 Transfer Matrix Method for Linear Real‐time Controlled Multibody Systems

9.5.1.1 Dynamic Model of Linear Controlled Multibody Systems

9.5.1.2 Transfer Equation and Extended Transfer Matrix of Elements

9.5.1.3 Overall Transfer Equation and Overall Transfer Matrix

9.5.1.4 Solving the Steady‐state Motion of the System

9.5.2 Transfer Matrix Method for Linear Delay Controlled Multibody Systems

9.6 Transfer Matrix Method for Controlled Multibody Systems II

9.6.1 Discrete Time Transfer Matrix Method of Real‐time Controlled Multibody Systems

9.6.2 Discrete Time Transfer Matrix Method for Controlled Multibody Systems with Time Delay

Table of Contents for
9 Transfer Matrix Method for Controlled Multibody Systems