Mobile Robot Dynamics
Mobile robot dynamics is a challenging field on its own, especially due to the variety of the imposed constraints. Delicate stability and control problems that have very often to be faced are due to longitudinal or lateral slip, and to the features of the ground (roughness, etc.). This chapter has the following objectives: (i) to present the general dynamic modeling concepts and techniques of robots, (ii) to study the Newton–Euler and Lagrange dynamic models of differential-drive mobile robots, (iii) to study the dynamics of differential-drive mobile robots with longitudinal and lateral slip, (iv) to derive a dynamic model of car-like wheeled mobile robots, (v) to derive a dynamic model of three-wheel omnidirectional robots, and (vi) to derive a dynamic model of four-wheel mecanum omnidirectional robots.
Keywords
Newton–Euler dynamic model; Lagrange dynamic model; slip dynamic model; force–torque diagram; car-like dynamic model; omnidirectional WMR dynamic model; three-wheel omnidirectional robot; four-wheel omnidirectional robot
3.1 Introduction
The next problem in the study of all types of robots, after kinematics, is the dynamic modeling [1–4]. Dynamic modeling is performed using the laws of mechanics that are based on the three physical elements: inertia, elasticity, and friction that are present in any real mechanical system such as the robot. Mobile robot dynamics is a challenging field on its own and has attracted considerable attention by researchers and engineers over the years. Most mobile robots, employed in practice, use conventional wheels and are subject to nonholonomic constraints that need particular treatment. Delicate stability and control problems, that often have to be faced in the design of a mobile robot, are due to the existence of longitudinal and lateral slip in the movement of the wheeled mobile robot (WMR) wheels [5–7].
This chapter has the following objectives:
• To present the general dynamic modeling concepts and techniques of robots.
• To study the Newton–Euler and Lagrange dynamic models of differential-drive mobile robots.
• To study the dynamics of differential-drive mobile robots with longitudinal and lateral slip.
• To derive a dynamic model of car-like WMRs.
• To derive a dynamic model of three-wheel omnidirectional robots.
• To derive a dynamic model of four-wheel mecanum omnidirectional robots.
3.2 General Robot Dynamic Modeling
Robot dynamic modeling deals with the derivation of the dynamic equations of the robot motion. This can be done using the following two methodologies:
The complexity of the Newton–Euler method is O(n), whereas the complexity of the Lagrange method can only be reduced up to 
, where n is the number of degrees of freedom.
Like kinematics, dynamics is distinguished in:
Direct dynamics provides the dynamic equations that describe the dynamic responses of the robot to given forces/torques 
that are exerted by the motors.
Inverse dynamics provides the forces/torques that are needed to get desired trajectories of the robot links. Direct and inverse dynamic modeling is pictorially illustrated in Figure 3.1.
In the inverse dynamic model the inputs are the desired trajectories of the link variables, and outputs the motor torques.
3.2.1 Newton–Euler Dynamic Model
This model is derived by direct application of the Newton–Euler equations for translational and rotational motion. Consider the object 
(robotic link, WMR, etc.) of Figure 3.2 to which a total force 
is applied at its center of gravity (COG).
and the inertial coordinate frame Oxyz.Then, its translational motion is described by:
(3.1)
Here, 
is the linear momentum given by:
(3.2)
where 
is the mass of the body and 
is the position of the COG with respect to the world (inertial) coordinate frame Oxyz. Assuming that 
is constant, then Eqs. (3.1) and (3.2) give:
(3.3)
which is the general translational dynamic model.
The rotational motion of 
is described by:
(3.4)
where 
is the total angular momentum of 
with respect to COG, and 
is the total external torque that produces the rotational motion of the body. The total momentum 
is given by:
(3.5)
Here, 
is the inertia tensor given by the volume integral:
(3.6)
where 
is the mass density of 
, dV is the volume of an infinitesimal element of 
lying at the position 
with respect to COG, 
is the angular velocity vector about the inertial axis passing from COG, 
is the 3×3 unit matrix, and 
is the volume of 
.
3.2.2 Lagrange Dynamic Model
The general Lagrange dynamic model of a solid body (multilink robot, WMR, etc.) is described by1:
(3.7)
where 
is the ith-degree of freedom variable, 
is the external generalized force vector applied to the body (i.e., force for translational motion, and torque for rotational motion), and L is the Lagrangian function defined by:
(3.8)
Here, K is the total kinetic energy, and P the total potential energy of the body given by:
(3.9)
where 
is the kinetic energy of link (degree of freedom) 
, and 
its potential energy. The kinetic energy K of the body is equal to:
(3.10)
where 
is the linear velocity of the COG, 
is the angular velocity of the rotation, m is the mass, and 
the inertia tensor of the body.
3.2.3 Lagrange Model of a Multilink Robot
Given a general multilink robot, the application of Eq. (3.7) leads (always) to a dynamic model of the form:
(3.11a)
where, for any 
is an 
positive define matrix, and:
(3.11b)
is the vector of generalized variables (linear, angular) 
represents the inertia force, 
represents the centrifugal and Coriolis force, and 
stands for the gravitational force. Since 
is the net force/torque applied to the robot, if there is also friction force/torque 
, then 
where 
is the force/torque exerted by the actuators at the joints.
It is useful to note that given the model (3.11a,b) one can express the kinetic energy of the robot as:
(3.12)
The expressions of 
, and 
have to be derived in each particular case. General derivations of Eq. (3.11a) from Eq. (3.7) are given in standard industrial (fixed) robotics books.
Typically, the function 
can be written in the form:
(3.13)
A useful universal property of the Lagrange model given by Eqs. (3.11a)–(3.13) is that the 
matrix 
is antisymmetric, that is, 
.
3.2.4 Dynamic Modeling of Nonholonomic Robots
The Lagrange dynamic model of a nonholonomic robot (fixed or mobile) has the form (3.7):
(3.14)
where 
is the 
matrix of the 
nonholonomic constraints:
(3.15)
and 
is the vector Lagrange multiplier. This model leads to:
(3.16)
where 
is a nonsingular transformation matrix. To eliminate the constraint term 
in Eq. (3.16) and get a constraint-free model we use an 
matrix 
which is defined such that:
(3.17)
From Eqs. (3.15) and (3.17), one can verify that there exists an 
-dimensional vector 
such that:
(3.18)
Now, premultiplying Eq. (3.16) by 
and using Eqs. (3.15), (3.17), and (3.18) we get:
(3.19a)
where:
(3.19b)
The reduced (unconstrained) model (3.19a,b) describes the dynamic evolution of the n-dimensional vector 
in terms of the dynamic evolution of the 
-dimensional vector 
.
3.3 Differential-Drive WMR
The dynamic model of the differential-drive WMR will be derived here by both the Newton–Euler and Lagrange methods.
3.3.1 Newton–Euler Dynamic Model
In the present case use will be made of the Newton–Euler equations:
(3.20a)
(3.20b)
where 
is the total force applied at the COG G, 
is the total torque with respect to the COG m is the mass of the WMR, and I is the inertia of the WMR. Referring to Figure 2.7, and assuming that the COG G coincides with the midpoint Q (i.e., b=0), we find:
(3.21a)
i.e:
(3.21b)
where 
and 
are the forces that produce the torques 
and 
, respectively.
Also:
(3.22)
Therefore, (3.20a,b) give the WMR’s dynamic model:
(3.23a)
(3.23b)
3.3.2 Lagrange Dynamic Model
We refer to Figure 2.7 and assume again that the point Q is at the position of point G. Here, the nonholonomic constraint matrix is (Eq. (2.42)):
(3.24)
Since the WMR moves on a horizontal planar terrain, the terms 
and 
in Eq. (3.16) are zero. Therefore, the model (3.16) becomes:
(3.25)
where:
(3.26a)
(3.26b)
To convert the model (3.25) to the corresponding unconstrained model (3.19a,b), we need the matrix 
in (3.17). Here, this matrix is:
(3.27)
which satisfies Eq. (3.17). Therefore, Eq. (3.19b) gives:
(3.28)
and the model (3.19a) becomes:
(3.29)
Noting that 
is the translation velocity v and 
the angular velocity 
of the robot, Eq. (3.29) gives the model:
(3.30a)
(3.30b)
which is identical to Eq. (3.23a,b), as expected. Finally, using Eq. (3.18) we get:
(3.31)
which is the WMR’s kinematic model. The dynamic and kinematic Eqs. (3.30a,b) and (3.31) describe fully the motion of the differential-drive WMR.
Example 3.1
The problem is to derive the Lagrange dynamic model using directly the Lagrangian function L for a differential-drive WMR in which:
1. There is linear friction in the wheels with the same friction coefficient.
Solution
We will work with the WMR of Figure 2.7.The kinetic energy K of the robot is given by:
(3.32)
where:
(3.33)
and (Example 2.2):
=mass of the entire robot
=linear velocity of the COG G
=moment of inertia of the robot with respect to Q
=moment of inertia of each wheel plus the corresponding motor’s rotor moment of inertia.
The velocities 
, and 
are given by Eq. (2.36a–c):
(3.34)
Using Eqs. (3.33) and (3.34) in Eq. (3.32) the total kinetic energy K of the robot is found to be:
(3.35)
Here the kinetic energy is expressed directly in terms of the angular velocities 
and 
of the driving wheels. The Lagrangian L is equal to K, since the robot is moving on a horizontal plane and so the potential energy P is zero. Therefore, the Lagrange dynamic equations of this robot are:
(3.36)
where β is the wheels’ common friction coefficient, and 
are the right and left actuation torques. Using (3.35) in (3.36) we get:
(3.37)
where:
(3.38)
, one can easily verify that in the case where the above conditions 1, 2, and 3 are relaxed (i.e., β=0, b=0, 
), the above dynamic model reduces to the model 3.3.3 Dynamics of WMR with Slip
Here we will derive the Newton–Euler dynamic model of the slipping differential-drive WMR considered in Example 2.2. The case where both longitudinal slip (variables 
) and lateral slip (variables 
) are present will be considered [5].
For convenience we write again the kinematic equations of the robot (with steering angles 
):
Defining the generalized variables’ vector 
as:
(3.39)
the above relations can be written in the following Pfaffian matrix form:
where:
(3.40)
The matrix 
and the velocity vector 
that satisfy the relations (Eqs. (3.17)–(3.18)):
(3.41)
(3.42)
where:
(3.43a)
(3.43b)
To derive the Newton–Euler dynamic model we draw the free-body diagram of the WMR body without the wheels (Figure 3.3), and the free-body diagram of the two wheels (Figure 3.4).
Figure 3.4 Force–torque diagram of the two wheels. The coordinate frame 
is fixed at the driving wheels’ midpoint.
In Figures 3.3 and 3.4, we have the following dynamic parameters and variables:
: The torque given to the wheels by the body of the WMR
: The driving torques exerted to the right and left wheel by their motors
: The mass of the WMR body without the wheels
: The mass of each driving wheel assembly (wheel plus DC motor of the robot body)
: The moment of inertia about a vertical axis passing via G (without the wheels)
: The wheel moment of inertia about its axis
: The wheel moment of inertia about its diameter
: The reaction forces between the WMR body and the wheels
: The lateral traction force at each wheel.
: The longitudinal traction force at each wheel.Using the above notation, the Newton–Euler dynamic equations of the robot, written down for each generalized variable in the vector (3.39), are:
(3.44)
where 
.
The detailed Eq. (3.44) can be written in the compact form of Eq. (3.16), namely:
(3.45)
where 
is given by (3.40), and:
Applying to Eq. (3.45) the procedure of Section 3.2.4, we get the following reduced model (3.19a,b):
(3.46)
where:
The model (3.46) can be split as:
(3.47a)
(3.47b)
(3.47c)
where:
(3.48)
with A, B, C, and D as in Eq. (3.43b). Writing 
in the form of Eq. (3.13), that is, 
, the model (3.47a) takes the form:
(3.49)
where:
Finally, Eq. (3.47b,c) can be written in the matrix form:
(3.50)
where:
(3.51)
To summarize, the dynamic model of the differential-drive WMR with slip is:
(3.52a)
(3.52b)
with:
(3.52c)
3.4 Car-Like WMR Dynamic Model
The kinematic equations of the car-like WMR were derived in Section 2.6. Here, we will derive the Newton–Euler dynamic model for a four-wheel rear-drive front-steer WMR [9]. The equivalent bicycle model shown in Figure 3.5 will be used.
The WMR is subject to the driving force 
and two lateral slip forces 
and 
applied perpendicular to the corresponding wheels. Before presenting the dynamic equations we derive the nonholonomic constraints that apply to the COG G which lies at a distance b from Q, and a distance d from P. To this end, we start with the nonholonomic constraints (2.50a,b) that refer to the points Q and P:
(3.53)
Denoting by 
and 
the coordinates of the point G in the world coordinate frame we get:
(3.54)
Introducing the relations (3.54) into (3.53) yields:
(3.55)
Now, denoting by 
the velocity of the COG G in the local coordinate frame 
and applying the rotational transformation (2.17) (Example 2.1) we get:
(3.56)
Introducing Eq. (3.56) into Eq. (3.55) yields:
(3.57)
which, upon differentiation, give:
(3.58a)
(3.58b)
From Eqs. (3.57) and (3.58a,b) we get:
(3.59)
Referring to Figure 3.5 we obtain the following Newton–Euler dynamic equations:
(3.60a)
(3.60b)
(3.60c)
(3.60d)
=mass of the WMR
=moment of inertia of the WMR about G
=driving force
=front and rear wheel lateral forces
=time constant of the steering system
=steering control input
=a constant coefficient (gain)
, with r the rear wheel radius, and 
the applied motor torque.2
We will now put the above dynamic equations into state space form, where the state vector is:
(3.61)
To this end, we solve Eq. (3.60c) for 
and introduce it, together with Eq. (3.58a) into (3.60b) to get:
(3.62)
Now introducing Eqs. (3.57), (3.58b), and (3.62) into Eq. (3.60a) we get:
(3.63a)
where:
(3.63b)
Finally, introducing Eq. (3.57) into Eq. (3.56) gives:
(3.64)
with 
. The model (3.64) represents an affine system with two inputs 
, a five-dimensional state vector:
(3.65a)
a drift term:
(3.65b)
and the two input fields:
(3.65c)
namely:
(3.66)
3.5 Three-Wheel Omnidirectional Mobile Robot
Here, we will derive the dynamic model of a three-wheel omnidirectional robot using the Newton–Euler method [10]. This derivation is the same for any number of universal (orthogonal) omniwheels. Some further studies of omnidirectional WMRs are presented in Refs. [11–14].
Consider the WMR in the pose shown in Figure 3.6 in which the three wheels are placed at angles 
, respectively.
The rotation matrix of the robot’s local coordinate frame 
with respect to the world coordinate frame 
is:
(3.67)
Let 
be the position vector of the COG 
. Then:
(3.68)
where 
, m is the robot’s mass, and 
is the force exerted at the COG expressed in the world coordinate frame. Denoting by:
(3.69)
the position vector of the COG and the force vector expressed in the local (moving) coordinate frame, then Eq. (3.67) implies that:
(3.70)
Thus, using Eq. (3.70) in Eq. (3.68) we get:
(3.71)
Now, the dynamic equation of the rotation about the COG Q is:
(3.72)
where 
is the moment of inertia of the robot about Q, and 
is the applied torque at Q.
From the geometry of Figure 3.6 we obtain:
(3.73)
where D is the distance of the wheels from the rotation point Q, and 
are the driving forces of the wheels.
The rotation of each wheel is described by the dynamic equation:
(3.74)
where 
is the common moment of inertia of the wheels, 
is the angular position of wheel 
, 
is the linear friction coefficient, 
is the common radius of the wheels, 
is the driving input torque of wheel 
, and 
is the driving torque gain.
Now, from the geometry of Figure 3.6 we see that the angles of the wheel velocities in the local coordinate frame are: 
Thus, we obtain the inverse kinematics equations from 
to 
as:
(3.75a)
(3.75b)
(3.75c)
Using Eqs. (3.69)–(3.75c) we get after some algebraic manipulation:
(3.76a)
(3.76b)
(3.76c)
Finally, combining Eqs. (3.67), (3.68), and (3.76a–c) we get the following state–space dynamic model of the WMR’s motion:
(3.77a)
(3.77b)
where:
The azimuth of the robot in the world coordinate frame is denoted by 
, where 
(
denotes the angle between 
and 
, that is, the azimuth of the robot in the moving coordinate frame). Then:
Whence:
(3.78)
where the positive direction is the counterclockwise rotation direction. It is noted that the motions along 
and 
are coupled because the dynamic equations are derived in the world coordinate frame. But since the rotational angle is always equal to 
, despite the fact that 
may be changing arbitrarily, the WMR can realize a translational motion without changing the pose (i.e., the WMR is holonomic). The model (3.77a) can be written in a three-input affine form with drift as follows:
(3.79)
where:
The dynamic model (3.77a) of a three-wheel omnidirectional robot (with universal wheels) is one of the many different models available. Actually, many other equivalent models were derived in the literature.
Example 3.2
Derive the dynamic equations of the three-wheel omnidirectional robot using the unit directional vectors 
of the wheel velocities.
Solution
To simplify the derivation we select the pose of the WMR in which the wheel 1 orientation is perpendicular to the local coordinate axis 
as shown in Figure 3.7 [15]. Thus the unit directional vectors 
, and 
are:
(3.80)
perpendicular to 
).The rotational matrix of 
with respect to 
is given by (3.67).
Therefore, the driving velocities 
of the wheels are:
or
(3.81)
where:
This is the inverse kinematic model of the robot. Now, applying the Newton–Euler method to the robot we get:
(3.82a)
(3.82b)
where 
is the magnitude of the driving force of the ith wheel, m is the robot mass, 
is the robot moment of inertia about Q, and:
(3.83)
are two-dimensional vectors found using Eqs. (3.67) and (3.80). The driving forces 
are given by the relation:
(3.84)
where 
is the voltage applied to the motor of the ith wheel, “a” is the voltage–force constant, and 
is the friction coefficient.
Combining Eqs. (3.82a,b), (3.83), and (3.84) we obtain the model:
(3.85a)
where:
(3.85b)
(3.85c)
The model (3.85a–c) has the standard form of the robot model described by Eqs. (3.11) and (3.13).
Example 3.3
We are given a car-like robot where there are lateral slip forces and longitudinal friction forces on all wheels. It is desired to write down the Newton–Euler dynamic equations in the form (3.60a–d).
Solution
The free-body diagram of the robot is as shown in Figure 3.8.
We use the following definitions:
where the upper index r refers to the right wheel and the upper index l to the left wheel. Therefore, considering the bicycle model of the WMR we get the following Newton–Euler dynamic equations in the local coordinate frame:
where all variables have the meaning presented in Section 3.4. From this point the development of the full model can be done as in Section 3.4.
3.6 Four Mecanum-Wheel Omnidirectional Robot
We consider the four-wheel omnidirectional robot of Figure 3.9A [3].
Figure 3.9 (A) The four-wheel mecanum WMR and the forces acting on it and (B) an experimental four-wheel mecanum WMR prototype. Source: www.robotics.ee.uwa.edu.au/eyebot/doc/robots/omni.html.
The total forces 
and 
acting on the robot in the x and y directions are:
(3.86a)
(3.86b)
where 
are the forces acting on the wheels along the x and y axes. In the absence of separate rotation motion, the direction of motion is defined by an angle 
where:
(3.87)
The torque 
that produces pure rotation is:
(3.88)
where positive rotation is in the counterclockwise direction.
The Newton–Euler motion equations are:
(3.89a)
(3.89b)
(3.89c)
where 
, and 
are the linear friction coefficients in the x, y, and 
motion, and 
are the mass and moment of inertia of the robot. Equations (3.89a–c) show that the robot can achieve steady-state velocities 
and 
equal to:
(3.90)
Example 3.4
The problem is to calculate the wheel angular velocities which are required to achieve a desired translational and rotational motion (WMR velocities v and 
) of the mecanum mobile robot of Figure 3.9A.
Solution
A solution to this problem was provided by the relations (2.84) and (2.85). Here we will provide an alternative method [3]. We draw the displacement and velocity vectors of a single wheel which are as shown in Figure 3.10A and B, where “
” is the roller angle 
.
Figure 3.10 (A) Displacement vectors of a wheel and roller and (B) velocity vectors (
is the rollers angle).
The vector 
represents the displacement due to the wheel rotation (in the positive direction), 
represents the displacement vector due to rolling which is orthogonal to the roller axis, and 
represents the total displacement vector. The dotted horizontal lines represent the discontinuities where the roller contact point transfers from one roller to the next. In Figure 3.10, the point A was selected in the middle of this discontinuity line to facilitate the calculation.
From Figure 3.10B we get 
because the components of 
and 
along the roller axis are equal. Therefore:
(3.91)
for 
. If 
, the rotation of the wheel does not produce translation motion of the rollers. Solving Eq. (3.91) for v we get:
(3.92)
When 
, the rotational speed 
of the wheel must be zero, but the wheel can have any value of translation velocity because of the motion of the other wheels.
We now calculate the wheel angular velocity 
for a desired translational motion velocity v. From Eq. (3.90), we see that the wheel velocity is proportional to the forces applied by each wheel, that is:
“
proportional to 
,” and because the WMR is a rigid body, all wheels should have the same translational speed, that is:
Therefore, from Eq. (3.92) we have:
(3.93)
Equation (3.93) gives the velocities 
of the four wheels needed to get a desired translational velocity v of the robot.
Finally, we will calculate the wheel angular velocities required to get a desired rotational speed 
. Consider a robot with velocity v. Then the instantaneous curvature radius (ICR) of its path is:
(3.94)
These relations give the following world frame coordinates 
of the ICR (Figure 3.11):
Figure 3.11 Geometry of wheel i with reference to the coordinate frame 
. Each wheel has its own total velocity 
and angle 
.
The geometry of each wheel is defined by its position 
and the orientation 
of its rollers. Let 
be the contact point of wheel i then:
(3.95)
where 
is the same for all wheels due to the symmetry of the wheels with respect to the point Q. The distance 
of the ICR and the contact point 
is given by the triangle formula, as:
(3.96)
The velocity 
of the wheel should be perpendicular to the line 
. Therefore, the angle 
of the line 
with the x-axis is determined by the relation:
(3.97)
whence:
(3.98)
because, in Figure 3.11, 
is actually negative 
. Now, in view of Eq. (3.94) we have:
(3.99)
Finally, using Eq. (3.99), Eq. (3.93) gives:
(3.100)
for 
, where r is the common radius of the wheels. Equation (3.100) gives the required rotational speeds of the wheels in terms of the position 
of ICR and the desired rotational speed 
of the robot. Actually, in view of Eqs. (3.94) and (3.96), Eq. (3.100) gives 
in terms of the known (desired) values of 
, 
, 
, and 
.
Example 3.5
It is desired to describe a way for identifying the dynamic parameters of a differential-drive WMR by converting its dynamic model to a linear form which uses the robot linear displacement in place of 
, and 
.
The nonlinear dynamic model of this WMR is given by Eqs. (3.30a,b) and (3.31) which is written as:
where 
and 
are the dynamic parameters to be identified. Using the robot’s linear displacement:
in place of the components 
and 
(Figure 2.7) we get the linear model:
which reduces to:
(3.101)
This is a linear model with two outputs 
and 
. The identification will be made rendering the model (3.101) to the standard linear regression model:
(3.102)
where:
The matrix 
is known as “regressor matrix.” Under the assumption that 
is independent of 
, and 
the solution for 
is given by3:
(3.103)
To convert Eq. (3.101) into the regression form (3.102) we discretize it in time using the first order approximation: 
, where 
and 
is the sampling period. Then, Eq. (3.101) becomes:
(3.104)
where 
. Clearly, the parameters 
and 
are identifiable, but the difficulty is that 
cannot be measured. To overcome this difficulty we use a second-order parametric representation of 
, and 
, namely [16]:
with boundary conditions:
From the above conditions, the parameters 
and 
are computed as:
(3.105a)
(3.105b)
The approximate length increment 
of 
is given by:
(3.106a)
where:
(3.106b)
Equation (3.106a) is an integral equation, the close solution of which is available in integral tables. The sign of 
, which determines whether the robot moved forward or backward, can be determined by the following relation:
where 
is the current position and 
is the past position of the robot. Several numerical experiments performed using the above WMR identification method gave very satisfactory results for both 
and 
[16].
3Equation (3.103) can be found by minimizing the function 
with respect to 
. An easy way to get Eq. (3.103) is by multiplying Eq. (3.102) by 
and solving for 
, under the assumption that 
(
independent of 
,and rank 
. Actually, Eq. (3.103) is 
where 
is the generalized inverse of 
(Eq. (2.8a)).
Example 3.6
Outline a method for applying the least squares identification model (3.102)–(3.103) to the general nonholonomic WMR model (3.19a–b)
Solution
is put in the form:
where 
is the vector of unknown parameters The above model can be written in the form (3.102) by computing 
as:
(3.107)
where 
and T is the sampling period of the measurements.
Now, defining 
and 
as:
The model (3.107), after N measurements, becomes:
(3.108)
This method overcomes the noise problem posed by the calculation of the acceleration. The optimal estimate 
of 
is given by Eq. (3.103):
(3.109)
The identification process can be simplified if it is split in two parts, namely: (i) identification when the robot is moving on a line without rotation, and (ii) identification when the robot has pure rotation movement. In the pure linear motion we keep the angle 
constant (i.e., 
), and in the pure rotation motion we have 
(i.e., 
and 
) [17].
References
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1Derivations of Eq. (3.7) from first principles are provided in textbooks on mechanics.
2Here, the driving motor dynamics is not included. Derivation of DC motor dynamic models is provided in Example 5.1.