• To present the fundamental analytical concepts required for the study of mobile robot kinematics

• To present the kinematic models of nonholonomic mobile robots (unicycle, differential drive, tricycle, and car-like wheeled mobile robots (WMRs))

• To present the kinematic models of 3-wheel, 4-wheel, and multiwheel omnidirectional WMRs.

• Direct and inverse robot kinematics

• Homogeneous transformations

• Nonholonomic constraints

2.2.1 Direct and Inverse Robot Kinematics

Consider a fixed or mobile robot with generalized coordinates in the joint (or actuation) space and in the task space. Define the vectors:

(2.1)

The problem of determining knowing is called the direct kinematics problem. In general and ( denotes the n-dimensional Euclidean space) are related by a nonlinear function (model) as:

(2.2)

The problem of solving Eq. (2.2), that is of finding from , is called the inverse kinematic problem expressed by:

(2.3)

The direct and inverse kinematic problems are pictorially shown in Figure 2.1.

In general, kinematics is the branch of mechanics that investigates the motion of material bodies without referring to their masses/moments of inertia and the forces/torques that produce the motion. Clearly, the kinematic equations depend on the fixed geometry of the robot in the fixed world coordinate frame.

To get these motions we must tune appropriately the motions of the joint variables, expressed by the velocities . We therefore need to find the differential relation of and . This is called direct differential kinematics and is expressed by:

(2.4)

where

and the matrix:

(2.5)

with element is called the Jacobian matrix of the robot.¹

For each configuration of the robot, the Jacobian matrix represents the relation of the displacements of the joints with the displacement of the position of the robot in the task space.

Let and be the velocities in the joint and task spaces.

Then, dividing Eq. (2.4) by we get formally:

(2.6)

Under the assumption that ( square) and that the inverse Jacobian matrix exists (i.e., its determinant is not zero: ), from Eq. (2.6) we get:

(2.7)

This is the inverse differential kinematics equation, and is illustrated in Figure 2.2.

If , then we have two cases:

Formally, the generalized inverse of a real matrix is defined to be the unique real matrix that satisfies the following four conditions:

It follows that has the properties:

All the above relations are useful when dealing with overspecified or underspecified linear algebraic systems (encountered, e.g., in underactuated or overactuated mechanical systems).

2.2.2 Homogeneous Transformations

The position and orientation of a solid body (e.g., a robotic link) with respect to the fixed world coordinate frame Oxyz (Figure 2.3) are given by a transformation matrix , called homogeneous transformation, of the type:

(2.9)

where is the position vector of the center of gravity (or some other fixed point of the link) with respect to Oxyz, and is a matrix defined as:

(2.10)

In Eq. (2.10), , and are the unit vectors along the axes , , of the local coordinate frame . The matrix represents the rotation of with respect to the reference (world) frame Oxyz. The columns , , and of are pairwise orthonormal, that is, where denotes the transpose (row) vector of the column vector , and denotes the Euclidean norm of , with , , and being the components of , respectively.

Thus the rotation matrix is orthonormal, that is:

(2.11)

To work with homogeneous matrices we use 4-dimensional vectors (called homogeneous vectors) of the type:

(2.12)

Suppose that and are the homogeneous position vectors of a point in the coordinate frames and Oxyz, respectively. Then, from Figure 2.3C we obtain the following vectorial equation:

where

Thus:

(2.13a)

or

(2.13b)

where is given by Eqs. (2.9) and (2.10). Equation (2.13b) indicates that the homogeneous matrix contains both the position and orientation of the local coordinate frame with respect to the world coordinate frame Oxyz.

It is easy to verify that:

(2.14)

Indeed, from Eq. (2.13a) we have: , which by Eq. (2.11) gives:

The columns , , and of consist of the direction cosines with respect to Oxyz. Thus the rotation matrices with respect to axes which are represented as:

are given by:

(2.15a)

(2.15b)

(2.15c)

where , , and are the rotation angles with respect to , , and , respectively.

In mobile robots moving on a horizontal plane, the robot is rotating only with respect to the vertical axis , and so Eq. (2.15c) is used. Thus, for convenience, we drop the index .

For better understanding, the upper left block of Eq. (2.15c) is obtained directly using the Oxy plane geometry shown in Figure 2.4.

Let a point in the coordinate frame Oxy, which is rotated about the axis Oz by the angle . The coordinates of in the frame are and as shown in Figure 2.4. From this figure we see that:

(2.16)

that is:

(2.17)

Similarly, one can derive the respective blocks and for the rotations about the and axes, respectively.

Given an open kinematic chain of links, the homogeneous vector of the local coordinate frame of the nth link, expressed in the world coordinate frame Oxyz can be found by successive application of Eq. (2.13b), that is, as:

(2.18)

where is the homogeneous transformation matrix that leads from the coordinate frame of link to that of link . The matrices can be computed by the so-called Denavit–Hartenberg (D–H) method (Section 10.2.1). The general relation (2.18) is of the form (2.2), and provides the robot Jacobian as indicated in Eq. (2.5).

2.2.3 Nonholonomic Constraints

A nonholonomic constraint (relation) is defined to be a constraint that contains time derivatives of generalized coordinates (variables) of a system and is not integrable. To understand what this means we first define a holonomic constraint as any constraint which can be expressed in the form:

(2.19)

where is the vector of generalized coordinates.

Now, suppose we have a constraint of the form:

(2.20)

If this constraint can be converted to the form:

(2.21)

we say that it is integrable. Therefore, although in Eq. (2.20) contains the time derivatives , it can be expressed in the holonomic form Eq. (2.21), and so it is actually a holonomic constraint. More specifically we have the following definition.

Typical systems that are subject to nonholonomic constraints (and hence are called nonholonomic systems) are underactuated robots, WMRs, autonomous underwater vehicles (AUVs), and unmanned aerial vehicles (UAVs). It is emphasized that “holonomic” does not necessarily mean unconstrained. Surely, a mobile robot with no constraint is holonomic. But a mobile robot capable of only translations is also holonomic.

Nonholonomicity occurs in several ways. For example a robot has only a few motors, say , where is the number of degrees of freedom, or the robot has redundant degrees of freedom. The robot can produce at most independent motions. The difference indicates the existence of nonholomicity. For example, a differential drive WMR has two controls (the torques of the two wheel motors), that is, , and three degrees of freedom, that is, . Therefore, it has one nonholonomic constraint.

In compact matrix form the above Pfaffian constraints can be written as:

(2.22)

An example of integrable Pfaffian constraint is:

(2.23)

This is integrable because it can be derived via differentiation, with respect to time, of the equation of a sphere:

with constant radius . The particular resulting sphere by integrating Eq. (2.23) depends on the initial state . The collection of all concentric spheres with center at the origin and radius “” is called a foliation with spherical leaves. For example, if the foliation produces a maximal integral manifold :

The nonholonomic constraint encountered in mobile robotics is the motion constraint of a disk that rolls on a plane without slipping (Figure 2.5). The no-slipping condition does not allow the generalized velocities , and to take arbitrary values.

Let be the disk radius. Due to the no-slipping condition the generalized coordinates are constrained by the following equations:

(2.24)

which are not integrable. These constraints express the condition that the velocity vector of the disk center lies in the midplane of the disk. Eliminating the velocity in Eq. (2.24) gives:

or

(2.25)

This is the nonholonomic constraint of the motion of the disk. Because of the kinematic constraints (2.24), the disk can attain any final configuration starting from any initial configuration . This can be done in two steps as follows:

Given a kinematic constraint one has to determine whether it is integrable or not. This can be done via the Frobenius theorem which uses the differential geometry concepts of distributions and Lie Brackets. We will come to this later (Section 6.2.1).

Two other systems that are subject to nonholonomic constraints are the rolling ball on a plane without spinning on place, and the flying airplane that cannot instantaneously stop in the air or move backward.

• Unicycle

• Differential drive WMR

• Tricycle WMR

• Car-like WMR

2.3.1 Unicycle

Unicycle has a kinematic model which is used as a basis for many types of nonholonomic WMRs. For this reason this model has attracted much theoretical attention by WMR controlists and nonlinear systems workers.

Unicycle is a conventional wheel rolling on a horizontal plane, while keeping its body vertical (Figure 2.5). The unicycle configuration (as seen from the bottom via a glass floor) is shown in Figure 2.6.

Its configuration is described by a vector of generalized coordinates: , that is, the position coordinates of the point of contact with the ground in the fixed coordinate frame Oxy, and its orientation angle with respect to the axis. The linear velocity of the wheel is and its angular velocity about its instantaneous rotational axis is . From Figure 2.6, we find:

(2.26)

Eliminating from the first two equations (2.26) we find the nonholonomic constraint (2.25):

(2.27)

Using the notation and , for simplicity, the kinematic model (2.26) of the unicycle can be written as:

(2.28a)

or

(2.28b)

where is the system Jacobian matrix:

(2.28c)

The linear velocity and the angular velocity are assumed to be the action (joint) variables of the system.

The model (2.28a) belongs to the special class of nonlinear systems, called affine systems, and described by a dynamic equation of the form (Chapter 6):

(2.29a)

(2.29b)

where appear linearly, and:

(2.30)

If the system has a less number of actuation variables (controls) than the degrees of freedom under control and is known as underactuated system. If we have an overactuated system. In practice, usually . The vector is actually the state vector of the system and the control vector. The term is called “drift,” and the system with is called a “driftless” system. The column vector set:

(2.31)

is referred to as the system’s vector field. It is assumed that the set contains at least an open set that involves the origin of . If does not contain the origin, then the system is not “driftless.”

The unicycle model (2.28a) is a 2-input driftless affine system with two vector fields:

(2.32)

The Jacobian formulation (2.28c) organizes the two column vector fields into a matrix . Each action variable in Eq. (2.29a) is actually a coefficient that determines how much of is contributing into the result . The vector field of the unicycle allows pure translation, and the field allows pure rotation.

2.3.2 Differential Drive WMR

Indoor and other mobile robots use the differential drive locomotion type (Figure 1.20). The Pioneer WMR shown in Figure 1.11 is an example of differential drive WMR. The geometry and kinematic parameters of this robot are shown in Figure 2.7. The pose (position/orientation) vector of the WMR and its speed are respectively:

(2.33)

The angular positions and speeds of the left and right wheels are , respectively.

The following assumptions are made:

Let and be the linear velocity of the left and right wheel respectively, and the velocity of the wheel midpoint of the WMR. Then, from Figure 2.7A we get:

(2.34a)

Adding and subtracting and v_l we get

(2.34b)

where, due to the nonslippage assumption, we have and . As in the unicycle case and are given by:

(2.35)

and so the kinematic model of this WMR is described by the following relations:

(2.36a)

(2.36b)

(2.36c)

Analogously to Eq. (2.28a,b) the kinematic model (2.36a–c) can be written in the driftless affine form:

(2.37a)

or

(2.37b)

where

(2.37c)

and is the WMR’s Jacobian:

(2.37d)

Here, the two 3-dimensional vector fields are:

(2.38)

The field allows the rotation of the right wheel, and allows the rotation of the left wheel. Eliminating in Eq. (2.35) we get as usual the nonholonomic constraint (2.25) or (2.27).

(2.39)

which expresses the fact that the point is moving along , and its velocity along the axis is zero (no lateral motion), that is (Figure 2.7B):

where and .

The Jacobian matrix in Eq. (2.37d) has three rows and two columns, and so it is not invertible. Therefore, the solution of Eq. (2.37b) for is given by:

(2.40)

where is the generalized inverse of given by Eq. (2.8a). However, here can be computed directly by using Eq. (2.34a), and observing from Figure 2.7B that:

Thus, using this equation in Eq. (2.34a) we obtain:

(2.41a)

that is:

or

(2.41b)

where³ :

(2.41c)

The nonholonomic constraint (2.39) can be written as:

(2.42)

Clearly, if , then the difference between and determines the robot’s rotation speed and its direction. The instantaneous curvature radius is given by (Eq. 1.1):

(2.43a)

and the instantaneous curvature coefficient is:

(2.43b)

Using these relations for and the above kinematic equations are written as:

and the nonholonomic constraint becomes:

In our WMR the two wheels have a common axis and are unsteered. Therefore, . For WMRs with steered wheels we may have , . In our case , and so the two kinematic equations, solved for the angular wheel velocities and , give:

where the inverse Jacobian is:

The nonholonomic constraints are written in Pfaffian form:

where

In the special case where only lateral slip takes place (i.e., , ), the components and are dropped from , and the matrices and are reduced appropriately, having only five columns. Note that here the wheels are fixed and so where is the lateral slipping velocity of the body of the WMR. Typically, the slipping variables, which are unknown and nonmeasurable are treated as disturbances via disturbance rejection and robust control techniques.

2.3.3 Tricycle

The motion of this WMR is controlled by the wheel steering angular velocity and its linear velocity (or its angular velocity , where is the radius of the wheel) (Figure 2.8).

The orientation angle and angular velocity are and , respectively. It is assumed that the vehicle has its guidance point in the back of the powered wheel (i.e., it has a central back axis). The state of the robot’s motion is:

The kinematic variables are:

Using the above relations we find:

(2.45)

where is the vector of joint velocities (control variables), and

(2.46)

is the Jacobian matrix. This Jacobian is again noninvertible, but we can find the inverse kinematic equations directly using the relations:

(2.47a)

and

(2.47b)

The instantaneous curvature radius is given by (Figure 2.8):

(2.48)

From Eq. (2.45) we see that the tricycle is again a 2-input driftless affine system with vector fields:

that allow steering wheel motion , and steering angle motion , respectively.

2.3.4 Car-Like WMR

The geometry of the car-like mobile robot is shown in Figure 2.9A and the A.W.E.S.O.M.-9000 line-tracking car-like robot prototype (Aalborg University) in Figure 2.9B.

The state of the robot’s motion is represented by the vector [20]:

(2.49)

where , are the Cartesian coordinates of the wheel axis midpoint , is the orientation angle of the vehicle, and is the steering angle. Here, we have two nonholonomic constraints, one for each wheel pair, that is:

(2.50a)

(2.50b)

where and are the position coordinates of the front wheels midpoint . From Figure 2.9 we get:

Using these relations the second kinematic constraint (2.50b) becomes:

The two nonholonomic constraints are written in the matrix form:

(2.51a)

where

(2.51b)

The kinematic equations for a rear-wheel driving car are found to be (Figure 2.9):

(2.52)

These equations can be written in the affine form:

(2.53)

that has the vector fields:

allowing the driving motion and the steering motion , respectively. The Jacobian form of Eq. (2.53) is:

(2.54)

with Jacobian matrix:

(2.55)

Here, there is a singularity at , which corresponds to the “jamming” of the WMR when the front wheels are normal to the longitudinal axis of its body. Actually, this singularity does not occur in practice due to the restricted range of the steering angle .

The kinematic model for the front wheel driving vehicle is Eqs. 2.45 and (2.46) [20]:

(2.56a)

In this case the previous singularity does not occur, since at the car can still (in principle) pivot about its rear wheels. Using the new inputs and defined as:

the above model is transformed to:

(2.56b)

where is the total steering angle with respect to the axis Ox.

Indeed, from and (Figure 2.9), and Eq. (2.56a) we get:

We observe, from Eq. (2.56b), that the kinematic model for , , and (i.e., the first, second, and fourth equation in Eq. (2.56b)) is actually a unicycle model (2.28a).

Two special cases of the above car-like model are known as:

The Reeds-Shepp car is obtained by restricting the values of the velocity to three distinct values , , and . These values appear to correspond to three distinct “gears”: “forward,” “park,” or “reverse.” The Dubins car is obtained when the reverse motion is not allowed in the Reeds-Shepp car, that is, the value is excluded, in which case .

2.3.5 Chain and Brockett—Integrator Models

The general 2-input n-dimensional chain model (briefly -chain model) is:

(2.58)

The Brockett (single) integrator model is:

(2.59)

and the double integrator model is:

(2.60)

The nonholonomic WMR kinematic models can be transformed to the above models. Here, the unicycle model (which also covers the differential drive model) and the car-like model will be considered.

2.3.6 Car-Pulling Trailer WMR

This is an extension of the car-like WMR, where one-axis trailers are attached to a car-like robot with rear-wheel drive. This type of trailer is used, for example, at airports for transporting luggage. The form of equations depend crucially on the exact point at which the trailer is attached and on the choice of body frames. Here, for simplicity each trailer will be assumed to be connected to the axle midpoint of the previous trailer (zero hooking) as shown in Figure 2.11 [20].

The new parameter introduced here is the distance from the center of the back axle of trailer to the point at which is hitched to the next body. This is called the hitch (or hinge-to-hinge) length denoted by . The car length is . Let be the orientation of the ith trailer, expressed with respect to the world coordinate frame. Then from the geometry of Figure 2.11 we get the following equations:

which give the following nonholonomic constraints:

for .

In analogy to Eq. (2.52) the kinematic equations of the N-trailer are found to be:

(2.70)

which, obviously, represent a driftless affine system with two inputs and , states:

We observe that the first four lines of the fields and represent the (powered) car-like WMR itself.

• Multiwheel omnidirectional WMR with orthogonal (universal) wheels

• Four-wheel omnidirectional WMR with mecanum wheels that have a roller angle .

2.4.1 Universal Multiwheel Omnidirectional WMR

The geometric structure of a multiwheel omnirobot is shown in Figure 2.12A. Each wheel has three velocity components [16]:

Here, the roller angle is , and so:

(2.71a)

where

(2.71b)

Thus the total velocity of the wheel is:

(2.72)

where and are the components of v_h, that is:

Equation (2.72) is general and can be used in WMRs with any number of wheels.

Thus, for example, in the case of a 3-wheel robot we may choose the angle for the wheels 1, 2, and 3 as 0°, 120°, and 240°, respectively, and get the equations:

(2.73)

with . Now, defining the vectors:

we can write Eq. (2.73) in the inverse Jacobian form:

(2.74a)

where

(2.74b)

Here , and Eq. (2.74a) can be inverted to give .

It is remarked that using omniwheels at different angles we can obtain an overall velocity of the WMR’s platform which is greater than the maximum angular velocity of each wheel. For example, selecting in the above 3-wheel case and we get from Eq. (2.71b):

The ratio is called the velocity augmentation factor (VAF) [16]:

and depends on the number of wheels used and their angular positions on the robot’s body. As a further example, consider a 4-wheel robot with and . Then, Eq. (2.71b) gives:

The relation between , and , is given by the rotational matrix (Eq. 2.17), that is:

or

Now, we have:

or, in compact, form:

(2.75a)

where

(2.75b)

(2.75c)

with:

(2.75d)

As usual, this inverse Jacobian equation gives the required angular wheel speeds that lead to the desired linear velocity , and angular velocity of the robot. A discussion of the modeling and control problem of a WMR with this structure is provided in Ref. [17].

2.4.2 Four–Wheel Omnidirectional WMR with Mecanum Wheels

Consider the 4-wheel WMR of Figure 2.14, where the mecanum wheels have roller angle [2,4].

Here, we have four-wheel coordinate frames . The angular velocity of the wheel has three components:

The wheel velocity vector in coordinates is given by:

(2.76)

for , where is the wheel radius, is the roller radius, and the roller angle. The robot velocity vector in the coordinate frame Eqs. 2.9–(2.13) is:

(2.77)

where denotes the rotation angle (orientation) of the frame with respect to , and , are the translations of with respect to . Introducing Eq. (2.76) into Eq. (2.77) we get:

(2.78)

where , and

(2.79)

is the Jacobian matrix of wheel , which is square and invertible. If all wheels are identical (except for the orientation of the rollers), the kinematic parameters of the robot in the configuration shown in Figure 2.14 are:

(2.80)

Thus, the Jacobian matrices (2.79) are:

(2.81)

The robot motion is produced by the simultaneous motion of all wheels.

In terms of (i.e., the wheels’ angular velocities around their axles) the velocity vector is given by:

(2.82)

The robot speed vector in the world coordinate frame is obtained as:

(2.83)

where is the rotation angle of the platform’s coordinate frame around the axis which is orthogonal to Oxy. Inverting Eqs. (2.82) and (2.83) we get the inverse kinematic model, which gives the angular speeds of the wheels around their hubs required to get a desired speed of the robot:

(2.84)

(2.85)

For historical awareness, we mention here that the mecanum wheel was invented by the Swedish engineer Bengt Ilon in 1973 during his work at the Swedish company Mecanum AB. For this reason it is also known as Ilon wheel or Swedish wheel.