2.2.1 Differential Drive

Differential drive is a very simple driving mechanism that is quite often used in practice, especially for smaller mobile robots. Robots with this drive usually have one or more castor wheels to support the vehicle and prevent tilting. Both main wheels are placed on a common axis. The velocity of each wheel is controlled by a separate motor. According to Fig. 2.3 the input (control) variables are the velocity of the right wheel v_R(t) and the velocity of the left wheel v_L(t).

The meanings of other variables in Fig. 2.3 are as follows: r is the wheel radius; L is the distance between the wheels and R(t) is the instantaneous radius of the vehicle driving trajectory (the distance between the vehicle center (middle point between the wheels) and ICR point). In each instance of time both wheels have the same angular velocity ω(t) around the ICR,

$\begin{array}{ll}\hfill \omega & =\frac{v_{\mathrm{L}}(t)}{R(t)-\frac{L}{2}}\hfill \\ \hfill \omega & =\frac{v_{\mathrm{R}}(t)}{R(t)+\frac{L}{2}}\hfill \\ \hfill \end{array}$

from where ω(t) and R(t) are expressed as follows:

$\begin{array}{ll}\hfill \omega (t)& =\frac{v_{\mathrm{R}}(t)-v_{\mathrm{L}}(t)}{L}\hfill \\ \hfill R(t)& =\frac{L}{2}\frac{v_{\mathrm{R}}(t)+v_{\mathrm{L}}(t)}{v_{\mathrm{R}}(t)-v_{\mathrm{L}}(t)}\hfill \\ \hfill \end{array}$

Tangential vehicle velocity is then calculated as

$\begin{array}{l}\hfill v(t)=\omega (t)R(t)=\frac{v_{\mathrm{R}}(t)+v_{\mathrm{L}}(t)}{2}\end{array}$

Wheel tangential velocities are v_L(t) = rω_L(t) and v_R(t) = rω_R(t), where ω_L(t) and ω_R(t) are left and right angular velocities of the wheels around their axes, respectively. Considering the above relations the internal robot kinematics (in local coordinates) can be expressed as

$\begin{array}{l}\hfill \left[\begin{array}{l}{\dot{x}}_{\mathrm{m}}(t)\\ {\dot{y}}_{\mathrm{m}}(t)\\ \dot{\phi }(t)\\ \end{array}\right]=\left[\begin{array}{l}{v}_{X_{\mathrm{m}}}(t)\\ v_{Y_{\mathrm{m}}}(t)\\ \omega (t)\\ \end{array}\right]=\left[\begin{array}{ll}\frac{r}{2}& \frac{r}{2}\\ 0& 0\\ -\frac{r}{L}& \frac{r}{L}\\ \end{array}\right]\left[\begin{array}{l}{\omega }_{\mathrm{L}}(t)\\ {\omega }_{\mathrm{R}}(t)\\ \end{array}\right]\end{array}$

Robot external kinematics (in global coordinates) is given by

$\begin{array}{l}\hfill \left[\begin{array}{l}\dot{x}(t)\\ \dot{y}(t)\\ \dot{\phi }(t)\\ \end{array}\right]=\left[\begin{array}{ll}cos(\phi (t))& 0\\ sin(\phi (t))& 0\\ 0& 1\\ \end{array}\right]\left[\begin{array}{l}v(t)\\ \omega (t)\\ \end{array}\right]\end{array}$

where v(t) and ω(t) are the control variables. Model (2.2) can be written in discrete form (2.3) using Euler integration and evaluated at discrete time instants t = kT_s, k = 0, 1, 2, … where T_s is the following sampling interval:

$\begin{array}{l}\hfill \begin{array}{cc}x(k+1)& =x(k)+v(k)T_{\mathrm{s}}cos(\phi (k))\\ y(k+1)& =y(k)+v(k)T_{\mathrm{s}}sin(\phi (k))\\ \phi (k+1)& =\phi (k)+\omega (k)T_{\mathrm{s}}\end{array}\end{array}$

2.2.2 Bicycle Drive

Bicycle drive is shown in Fig. 2.4. It has a steering wheel where α is the steering angle and ω_S is wheel angular velocity around its axis (front-wheel drive). The ICR point is defined by the intersection of both wheel axes. In each moment of time the bicycle circles around ICR with angular velocity ω, radius R, and distance between the wheels d:

$\begin{array}{l}\hfill R(t)=dtan\left(\frac{\pi }{2}-\alpha (t)\right)=\frac{d}{tan\left(\alpha (t)\right)}\end{array}$

The steering wheel circles around ICR with ω so we can write

$\begin{array}{l}\hfill \omega (t)=\dot{\phi }=\frac{v_{\mathrm{s}}(t)}{\sqrt{d^2+R^2}}=\frac{v_{\mathrm{s}}(t)}{d}sin\left(\alpha (t)\right)\end{array}$

where v_s(t) = ω_s(t)r and r are the rim velocity and the radius of the front wheel, respectively.

Internal robot kinematics (in robot frame) is

$\begin{array}{l}\hfill \begin{array}{cc}{\dot{x}}_{\mathrm{m}}& =v_{\mathrm{s}}(t)cos\left(\alpha (t)\right)\\ {\dot{y}}_{\mathrm{m}}& =0\\ \dot{\phi }& =\frac{v_{\mathrm{s}}(t)}{d}sin\left(\alpha (t)\right)\end{array}\end{array}$

and the external kinematic model is

$\begin{array}{l}\hfill \begin{array}{cc}l\dot{x}& =v_{\mathrm{s}}(t)cos\left(\alpha (t)\right)cos\left(\phi (t)\right)\\ \dot{y}& =v_{\mathrm{s}}(t)cos\left(\alpha (t)\right)sin\left(\phi (t)\right)\\ \dot{\phi }& =\frac{v_{\mathrm{s}}(t)}{d}sin\left(\alpha (t)\right)\end{array}\end{array}$

or in compact form

$\begin{array}{l}\hfill \left[\begin{array}{l}\dot{x}\\ \dot{y}\\ \dot{\phi }\\ \end{array}\right]=\left[\begin{array}{ll}cos\left(\phi (t)\right)& 0\\ sin\left(\phi (t)\right)& 0\\ 0& 1\\ \end{array}\right]\left[\begin{array}{l}v(t)\\ \omega (t)\\ \end{array}\right]\end{array}$

where $v=v_{\mathrm{s}}(t)cos\left(\alpha (t)\right)$ and $\omega (t)=\frac{v_{\mathrm{s}}(t)}{d}sin\left(\alpha (t)\right)$.

2.2.3 Tricycle Drive

Tricycle drive (see Fig. 2.5) has the same kinematics as bicycle drive.

$\begin{array}{l}\hfill \begin{array}{cc}\dot{x}& =v_{\mathrm{s}}(t)cos\left(\alpha (t)\right)cos\left(\phi (t)\right)\\ \dot{y}& =v_{\mathrm{s}}(t)cos\left(\alpha (t)\right)sin\left(\phi (t)\right)\\ \dot{\phi }& =\frac{v_{\mathrm{s}}(t)}{d}sin\left(\alpha (t)\right)\end{array}\end{array}$

where $v=v_{\mathrm{s}}(t)cos\left(\alpha (t)\right)$, $\omega (t)=\frac{v_{\mathrm{s}}(t)}{d}sin\left(\alpha (t)\right)$, and v_s is the rim velocity of the steering wheel.

Tricycle drive is more common in mobile robotics because of the three wheels that make the robot stable in a vertical direction by itself.

2.2.4 Tricycle With a Trailer

The tricycle part of kinematics is already described in Section 2.2.3. For a trailer ICR₂ is found by the intersection of the tricycle and the trailer wheel axes. The angular velocity at which the trailer wheels circle around the ICR₂ point is

$\begin{array}{l}\hfill {\omega }_2(t)=\frac{v}{R_2}=\frac{v_{\mathrm{s}}cos\alpha }{R_2}=\frac{v_{\mathrm{s}}cos\alpha sin\beta }{L}=\dot{\beta }\end{array}$

The final kinematic model (for Fig. 2.6) is then obtained as follows:

$\begin{array}{l}\hfill \begin{array}{cc}\dot{x}& =v_{\mathrm{s}}(t)cos\left(\alpha (t)\right)cos\left(\phi (t)\right)\\ \dot{y}& =v_{\mathrm{s}}(t)cos\left(\alpha (t)\right)sin\left(\phi (t)\right)\\ \dot{\phi }& =\frac{v_{\mathrm{s}}(t)}{d}sin\left(\alpha (t)\right)\\ \dot{\beta }& =\frac{v_{\mathrm{s}}cos\alpha sin\beta }{L}\end{array}\end{array}$

2.2.5 Car (Ackermann) Drive

Cars use the Ackermann steering principle. The idea behind the Ackermann steering is that the inner wheel (closer to ICR) should steer for a bigger angle than the outer wheel in order to allow the vehicle to rotate around the middle point between the rear wheel axis. Consequently the inner wheel travels with a slower speed than the outer wheel. The Ackermann driving mechanism allows for the rear wheels to have no slip angle, which requires that the ICR point lies on a straight line defined by the rear wheels’ axis. This driving mechanism therefore minimizes tire wear.

In Fig. 2.7 the left wheel is on the outer side and the right wheel is on the inner side. The steering orientations of the front wheels are therefore determined from

$\begin{array}{l}\hfill \begin{array}{cc}tan\left(\frac{\pi }{2}-{\alpha }_L\right)& =\frac{R+\frac{l}{2}}{d}\\ tan\left(\frac{\pi }{2}-{\alpha }_R\right)& =\frac{R-\frac{l}{2}}{d}\end{array}\end{array}$

and the final steering angles are expressed as

$\begin{array}{l}\hfill \begin{array}{cc}{\alpha }_L& =\frac{\pi }{2}-arctan\frac{R+\frac{l}{2}}{d}\\ {\alpha }_R& =\frac{\pi }{2}-arctan\frac{R-\frac{l}{2}}{d}\end{array}\end{array}$

The inner and outer back wheels circle around the ICR with the same angular velocity ω; therefore their rim velocities are

$\begin{array}{l}\hfill \begin{array}{cc}l{v}_L& =\omega \left(R+\frac{l}{2}\right)\\ v_R& =\omega \left(R-\frac{l}{2}\right)\end{array}\end{array}$

This type of kinematic drive is especially appropriate to model larger vehicles. A motion model can also be described using tricycle kinematics (Eq. 2.23) where the average the Ackermann angle $\alpha =\frac{\pi }{2}-arctan\frac{R}{d}$ is used.

Inverse kinematics of Ackermann drive is complicated and exceeds the scope of this work.

2.2.6 Synchronous Drive

If a vehicle is able to steer each of its wheels around the vertical axis synchronously (wheels have the same orientation and are steered synchronously) then this is called synchronous drive (also synchro drive). Typical synchro drive has three wheels arranged symmetrically (equilateral triangle) around the vehicle center as in Fig. 2.8.

All wheels are synchronously steered so their rotation axes are always parallel and the ICR point is at infinity. The vehicle can directly control the orientation of the wheels, which presents the third state in the state vector (2.27). The control variables are wheels’ steering velocity ω, and forward speed v.

The kinematics has a similar form as in the case of differential drive:

$\begin{array}{l}\hfill \left[\begin{array}{l}\dot{x}(t)\\ \dot{y}(t)\\ \dot{\phi }(t)\\ \end{array}\right]=\left[\begin{array}{ll}cos(\phi (t))& 0\\ sin(\phi (t))& 0\\ 0& 1\\ \end{array}\right]\left[\begin{array}{l}v(t)\\ \omega (t)\\ \end{array}\right]\end{array}$

where v(t) and ω(t) (steering velocity of the wheels) are the control variables that can be controlled independently (which is not the case for differential drive).

2.2.7 Omnidirectional Drive

In previous kinematic models simple wheels were used where the wheel can only move (roll) in the direction of its orientation. Such simple wheels have only one rolling direction. To allow omnidirectional rolling motion (in more directions) a complex wheel construction is required. An example of such a wheel is the Mecanum wheel (see Fig. 2.9) or Swedish wheel where the number of smaller rollers are arranged around the main wheel rim. The passive roller axes are not aligned with the main wheel axis (typically they are at a γ = 45 degree angle). This allows a number of different motion directions that result from an arbitrary combination of the main wheel’s rotation direction and passive rollers’ rotation.

Kinematics of Four-Wheel Omnidirectional Drive

Another example of a complex wheel is the omni wheel or poly wheel, which enables omnidirectional motion similarly to the Mecanum wheel. The omni wheel (Fig. 2.10) has passive rollers mounted around the wheel rim so that their axes are 90 degree to the wheel axis.

The popular four-wheel platform known as the Mecanum platform is shown in Fig. 2.11. The Mecanum platform has wheels with left- and right-handed rollers where diagonal wheels are of the same type. This enables the vehicle to move in any direction with arbitrary rotation by commanding a proper velocity to each main wheel. Forward or backward motion is obtained by setting all main wheels to the same velocity. If velocities of the main wheels on one side are opposite to the velocities of the wheels on the other side then the platform will rotate. Side motion of the platform is obtained if the wheels on one diagonal have opposite velocity to the wheels on the other diagonal. With combination of the motions described, motion of the platform in any direction and with any rotation can be achieved.

The inverse internal kinematics of the four-wheel Mecanum drive in Fig. 2.12 can be derived as follows.

The first wheel velocity in robot coordinates is obtained from the main wheel velocity v₁(t) and the passive roller’s velocity v_R(t). In the sequel the time dependency notation is omitted to have more compact and easy-to-follow equations (e.g., v₁(t) = v₁). The total wheel velocity in x_m and y_m direction of the robot coordinate frame are $v_{\mathrm{m}1x}=v_1+v_{\mathrm{R}}cos(\frac{\pi }{4})=v_1+\frac{v_{\mathrm{R}}}{\sqrt{2}}$ and $v_{\mathrm{m}1y}=v_{\mathrm{R}}sin(\frac{\pi }{4})=\frac{v_{\mathrm{R}}}{\sqrt{2}}$, from where the main wheel velocity is obtained: v₁ = v_m1x − v_m1y. The first wheel velocity in the robot coordinate frame direction can also be expressed with the robot’s translational velocity $v_{\mathrm{m}}=\sqrt{{\dot{x}}_{\mathrm{m}}^2+{\dot{y}}_{\mathrm{m}}^2}$ and its angular velocity $\dot{\phi }$ as follows: $v_{\mathrm{m}1x}={\dot{x}}_{\mathrm{m}}-\dot{\phi }d$ and $v_{\mathrm{m}1y}={\dot{y}}_{\mathrm{m}}+\dot{\phi }l$ (the meaning of distances d and l can be deduced from Fig. 2.12). From the latter relations the main wheel velocity can be expressed by the robot body velocity as $v_1={\dot{x}}_{\mathrm{m}}-{\dot{y}}_{\mathrm{m}}-(l+d)\dot{\phi }$. Similar equations can be obtained for v₂, v₃, and v₄. The inverse kinematics in local coordinates is

$\begin{array}{l}\hfill \left[\begin{array}{l}{v}_1\\ v_2\\ v_3\\ v_4\\ \end{array}\right]=\left[\begin{array}{lll}1& -1& -(l+d)\\ 1& 1& -(l+d)\\ 1& -1& (l+d)\\ 1& 1& (l+d)\\ \end{array}\right]\left[\begin{array}{l}{\dot{x}}_{\mathrm{m}}\\ {\dot{y}}_{\mathrm{m}}\\ \dot{\phi }\\ \end{array}\right]\end{array}$

  (2.29)

The internal inverse kinematics (Eq. 2.29) in compact form reads $\mathit{v}=\mathit{J}{\dot{\mathit{q}}}_{\mathrm{m}}$, where v^T = [v₁, v₂, v₃, v₄]^T and q_m^T = [x_m, y_m, φ]^T.

To calculate the inverse kinematics in global coordinates the rotation matrix representing the orientation of the local coordinates with respect to the global coordinates (${\mathit{q}}_{\mathrm{m}}={\mathit{R}}_G^L\mathit{q}$),

$\begin{array}{l}\hfill {\mathit{R}}_G^L=\left[\begin{array}{lll}cos\phi & sin\phi & 0\\ -sin\phi & cos\phi & 0\\ 0& 0& 1\\ \end{array}\right]\end{array}$

needs to be considered as follows: $\mathit{v}=\mathit{J}{\mathit{R}}_G^L\dot{\mathit{q}}$.

From the internal inverse kinematics $\mathit{v}=\mathit{J}{\dot{\mathit{q}}}_{\mathrm{m}}$ (Eq. 2.29), the forward internal kinematics is obtained by ${\dot{\mathit{q}}}_{\mathrm{m}}={\mathit{J}}^{+}\mathit{v}$, where ${\mathit{J}}^{+}={\left({\mathit{J}}^T\mathit{J}\right)}^{-1}{\mathit{J}}^T$ is the pseudo-inverse of J. Forward internal kinematics for the four-wheel Mecanum platform reads as follows:

$\begin{array}{l}\hfill \left[\begin{array}{l}{\dot{x}}_{\mathrm{m}}\\ {\dot{y}}_{\mathrm{m}}\\ \dot{\phi }\\ \end{array}\right]=\frac{1}{4}\left[\begin{array}{llll}1& 1& 1& 1\\ -1& 1& -1& 1\\ \frac{-1}{(l+d)}& \frac{-1}{(l+d)}& \frac{1}{(l+d)}& \frac{1}{(l+d)}\\ \end{array}\right]\left[\begin{array}{l}{v}_1\\ v_2\\ v_3\\ v_4\\ \end{array}\right]\end{array}$

  (2.30)

Forward kinematics in global coordinates is obtained by $\dot{\mathit{q}}={({\mathit{R}}_G^L)}^T{\mathit{J}}^{+}\mathit{v}$.

Kinematics of Three-Wheel Omnidirectional Drive

A popular omnidirectional configuration for three-wheel drive is shown in Fig. 2.13.

Its inverse kinematics (in global coordinates) is obtained by considering the robot translational velocity $v=\sqrt{{\dot{x}}^2+{\dot{y}}^2}$ and its angular velocity $\dot{\phi }$. Velocity of the first wheel v₁ = v_1t + v_1r consists of a translational part $v_{1t}=-\dot{x}sin(\phi )+\dot{y}cos(\phi )$ and an angular part $v_{1r}=R\dot{\phi }$. The first wheel common velocity therefore is $v_1=-\dot{x}sin(\phi )+\dot{y}cos(\phi )+R\dot{\phi }$. Similarly, considering the global angle of the second wheel φ + θ₂, its velocity is $v_2=-\dot{x}sin(\phi +{\theta }_2)+\dot{y}cos(\phi +{\theta }_2)+R\dot{\phi }$. And the velocity of the third wheel reads $v_3=-\dot{x}sin(\phi +{\theta }_3)+\dot{y}cos(\phi +{\theta }_3)+R\dot{\phi }$. The inverse kinematics in global coordinates of three-wheel drive is

$\begin{array}{l}\hfill \left[\begin{array}{l}{v}_1\\ v_2\\ v_3\\ \end{array}\right]=\left[\begin{array}{lll}-sin(\phi )& cos(\phi )& R\\ -sin(\phi +{\theta }_2)& cos(\phi +{\theta }_2)& R\\ -sin(\phi +{\theta }_3)& cos(\phi +{\theta }_3)& R\\ \end{array}\right]\left[\begin{array}{l}\dot{x}\\ \dot{y}\\ \dot{\phi }\\ \end{array}\right]\end{array}$

  (2.31)

The inverse kinematics (Eq. 2.31) in compact form reads $\mathit{v}=\mathit{J}\dot{\mathit{q}}$. Sometimes it is more convenient to steer the robot in its local coordinates, which can be obtained by considering rotation transformation as follows: $\mathit{v}=\mathit{J}{({\mathit{R}}_G^L)}^T{\dot{\mathit{q}}}_{\mathrm{m}}$.

Forward kinematics in global coordinates is obtained from the inverse kinematics (Eq. 2.31) by $\dot{\mathit{q}}=\mathit{S}\mathit{v}$ where S =J⁻¹:

$\begin{array}{l}\hfill \left[\begin{array}{l}\dot{x}\\ \dot{y}\\ \dot{\phi }\\ \end{array}\right]=\frac{2}{3}\left[\begin{array}{lll}-sin({\theta }_1)& -sin({\theta }_1+{\theta }_2)& -sin({\theta }_1+{\theta }_3)\\ cos({\theta }_1)& cos({\theta }_1+{\theta }_2)& cos({\theta }_1+{\theta }_3)\\ \frac{1}{2R}& \frac{1}{2R}& \frac{1}{2R}\\ \end{array}\right]\left[\begin{array}{l}{v}_1\\ v_2\\ v_3\\ \end{array}\right]\end{array}$

  (2.32)

2.2.8 Tracked Drive

The kinematics of the tracked drive (Fig. 2.14) can be approximately described by the differential drive kinematics.

$\begin{array}{l}\hfill \left[\begin{array}{l}\dot{x}(t)\\ \dot{y}(t)\\ \dot{\phi }(t)\\ \end{array}\right]=\left[\begin{array}{ll}cos(\phi (t))& 0\\ sin(\phi (t))& 0\\ 0& 1\\ \end{array}\right]\left[\begin{array}{l}v(t)\\ \omega (t)\\ \end{array}\right]\end{array}$

However, the differential drive assumes perfect rolling contact between its wheels and the ground, which is not the case for tracked drive. The tracked drive (Caterpillar tractor) has a bigger contact surface between its wheels and the ground and requires wheels slipping to change its direction. The tracked drives can therefore move through more rough terrain where wheeled vehicles usually cannot.

The amount of sleep between the Caterpillar tractor and the ground is not constant and depends on ground contact; therefore the odometry (direct kinematics) is even less reliable to estimate the robot pose compared to the differential drive.

2.3.1 Holonomic Constraints

These constraints depend on generalized coordinates. For a system with n generalized coordinates q = [q₁, …, q_n]^T a holonomic constraint is expressed in the following form:

$\begin{array}{l}\hfill f(\mathit{q})=f(q_1,\dots ,q_n)=0\end{array}$

where f and its derivatives are continuous functions. This constraint defines a subspace of all possible configurations in generalized coordinates for which Eq. (2.34) is true. Constraint (2.34) can be used to eliminate certain generalized coordinates (it can be expressed by n − 1 other coordinates).

In general we can have m holonomic constraints (m < n). If these constraints are linearly independent then they define (n − m)-dimensional subspace, which is true configuration space (work space, system has n − m DOFs).

2.3.2 Nonholonomic Constraints

Nonholonomic constraints limit possible velocities of the system or possible directions of motion. The nonholonomic constraint can be formulated by

$\begin{array}{l}\hfill f(\mathit{q},\dot{\mathit{q}})=f(q_1,\dots ,q_n,{\dot{q}}_1,\dots ,{\dot{q}}_n)=0\end{array}$

where f is a smooth function with continuous derivatives and $\dot{\mathit{q}}$ is the vector of system velocities in the generalized coordinates. In case the system has no constraints (2.35), it has no limitation in motion directions.

A kinematic constraint (2.35) is holonomic if it is integrable, which means that velocities ${\dot{q}}_1,\dots ,{\dot{q}}_n$ can be eliminated from Eq. (2.35) and the constraint can be expressed in the form of Eq. (2.34). If the constraint (2.35) is not integrable, then it is nonholonomic.

If m linearly independent nonholonomic constraints exist in the form Eq. (2.35), the velocity space is (n − m)-dimensional. Nonholonomic constraints limit allowable velocities of the system. For example, a differential drive vehicle (e.g., wheelchair) can move in the direction of the current wheel orientation but not in the lateral direction.

Assuming linear constraints in $\dot{\mathit{q}}={[{\dot{q}}_1,\dots ,{\dot{q}}_n]}^T$, Eq. (2.35) can be written as

$\begin{array}{l}\hfill f(\mathit{q},\dot{\mathit{q}})={\mathit{a}}^T(\mathit{q})\dot{\mathit{q}}=\left[\begin{array}{lll}{a}_1(\mathit{q})& \cdots & a_n(\mathit{q})\\ \end{array}\right]\left[\begin{array}{l}{\dot{q}}_1\\ ⋮\\ {\dot{q}}_n\\ \end{array}\right]=0\end{array}$

where a(q) is a parameter vector of the constraint. For m nonholonomic constraints a constraint matrix is obtained as follows:

$\begin{array}{l}\hfill \mathit{A}(\mathit{q})=\left[\begin{array}{l}{\mathit{a}}_1^T(\mathit{q})\\ ⋮\\ {\mathit{a}}_{\mathrm{m}}^T(\mathit{q})\\ \end{array}\right]\end{array}$

and all nonholonomic constraints are given in matrix form

$\begin{array}{l}\hfill \mathit{A}(\mathit{q})\dot{\mathit{q}}=0\end{array}$

In each time instant a matrix of reachable motion directions is S(q) = [s₁(q), …, s_n−m(q)] (m constraints define (n − m) reachable directions). This matrix defines the kinematic model as follows:

$\begin{array}{l}\hfill \dot{\mathit{q}}(t)=\mathit{S}(\mathit{q})\mathit{v}(t)\end{array}$

where v(t) is the control vector (see kinematic model (2.2)). The product of the constraint matrix A and the kinematic matrix S is a zero matrix:

$\begin{array}{l}\hfill \mathit{A}\mathit{S}=0\end{array}$

2.3.3 Integrability of Constraints

To determine if some constraint is nonholonomic, an integrability test needs to be done. If it cannot be integrated to obtain the form of holonomic constraint, the constraint is nonholonomic.

2.3.4 Vector Fields, Distribution, Lie Bracket

In current time t and current state q all possible motion directions can be obtained by a linear combination of vector fields in matrix S. Distribution is defined as a reachable subspace from q using motions defined by a linear combination of vector fields (columns in S).

Vector fields are time derivatives of generalized coordinates and therefore represent velocities or motion directions. A vector field is a mapping that is continuously differentiable and assigns a specific vector to each point in state space. A demonstration of reachable vector field determination for differential drive is given in Example 2.1.

If distribution of some vector fields span the whole space then it is involutive. For noninvolutive distribution of some vector fields a set of new vector fields (linearly independent from vector fields of the basic distribution) can be defined. These new vector fields (motion directions) can be obtained by a finite switching of the basic motion directions with movement length (in each basic direction) limited to zero. New motion directions can be calculated using Lie brackets, which operates over two vector fields as shown later in Eq. (2.44). Parallel parking of a car or differential drive is a practical example of obtaining a new motion direction by switching between the basic motion directions. The car cannot directly move in the lateral direction to the parking slot. However, it can still go into a parking slot by lateral direction movement from a series of forward, backward, and rotation motions as illustrated in Example 2.2.

Example 2.2

Illustration of parallel parking maneuver of differential drive vehicle.

Solution

Using the basic motion directions defined by the vector fields s₁(q) and s₂(q) (see Eq. 2.41) and the initial vehicle pose q₀ = q(0) new vehicle poses are obtained as follows. Starting from q₀ = q(0) and driving in the direction s₁ for a short period of time ε, then driving in the direction s₂ for time ε followed by driving in the direction −s₁ for time ε, and finally for time ε in the direction −s₂, the final pose is obtained. Mathematically this is formulated as follows:

$\begin{array}{l}\hfill \mathit{q}(4\epsilon )={\varphi }_{\epsilon }^{-{\mathit{s}}_2}\left({\varphi }_{\epsilon }^{-{\mathit{s}}_1}\left({\varphi }_{\epsilon }^{{\mathit{s}}_2}\left({\varphi }_{\epsilon }^{{\mathit{s}}_1}\left({\mathit{q}}_0\right)\right)\right)\right)\end{array}$

which represents a nonlinear differential equation whose solution can be approximated by Taylor series expansion (see details in [1]) by

$\begin{array}{l}\hfill \mathit{q}(4\epsilon )={\mathit{q}}_0+{\epsilon }^2\left(\frac{\partial {\mathit{s}}_2}{\partial \mathit{q}}{\mathit{s}}_1({\mathit{q}}_0)-\frac{\partial {\mathit{s}}_1}{\partial \mathit{q}}{\mathit{s}}_2({\mathit{q}}_0)\right)+O({\epsilon }^3)\end{array}$

  (2.43)

where partial derivatives are estimated for q₀ and O(ε³) is the remaining part belonging to a higher order of derivatives that can be ignored for a short time ε. The obtained final motion distance of the parallel parking maneuver is ε² in the direction of the vector obtained by Lie brackets (defined in Eq. 2.44).

Illustration of this maneuver can be done by the following experiment. Suppose the starting vehicle pose is q₀ = [0, 0, 0]^T. By performing the first step of the parking maneuver the resulting pose is

$\begin{array}{l}\hfill {\mathit{q}}_1={\mathit{q}}_0+\epsilon {\mathit{s}}_1=\left[\begin{array}{l}0\\ 0\\ 0\\ \end{array}\right]+\epsilon \left[\begin{array}{l}cos0\\ sin0\\ 0\\ \end{array}\right]=\left[\begin{array}{l}\epsilon \\ 0\\ 0\\ \end{array}\right]\end{array}$

In the second step a rotation is performed

$\begin{array}{l}\hfill {\mathit{q}}_2={\mathit{q}}_1+\epsilon {\mathit{s}}_2=\left[\begin{array}{l}\epsilon \\ 0\\ 0\\ \end{array}\right]+\epsilon \left[\begin{array}{l}0\\ 0\\ 1\\ \end{array}\right]=\left[\begin{array}{l}\epsilon \\ 0\\ \epsilon \\ \end{array}\right]\end{array}$

The third step contains translation in the negative direction:

$\begin{array}{l}\hfill {\mathit{q}}_3={\mathit{q}}_2-\epsilon {\mathit{s}}_1=\left[\begin{array}{l}\epsilon \\ 0\\ \epsilon \\ \end{array}\right]-\epsilon \left[\begin{array}{l}cos\epsilon \\ sin\epsilon \\ 0\\ \end{array}\right]=\left[\begin{array}{l}\epsilon -\epsilon cos\epsilon \\ -\epsilon sin\epsilon \\ \epsilon \\ \end{array}\right]\end{array}$

and in the last step there is a rotation in the negative direction:

$\begin{array}{l}\hfill {\mathit{q}}_4={\mathit{q}}_3-\epsilon {\mathit{s}}_2=\left[\begin{array}{l}\epsilon -\epsilon cos\epsilon \\ -\epsilon sin\epsilon \\ \epsilon \\ \end{array}\right]-\epsilon \left[\begin{array}{l}0\\ 0\\ 1\\ \end{array}\right]=\left[\begin{array}{l}\epsilon -\epsilon cos\epsilon \\ -\epsilon sin\epsilon \\ 0\\ \end{array}\right]\end{array}$

Computed motions and intermediate points are shown in Fig. 2.15. The final pose of this maneuver is in the lateral direction to the starting pose. The final pose, however, is not directly accessible with basic motion vector fields s₁ and s₂, but it can be achieved by a combination of the basic vector fields. The obtained final motion is not strictly in the lateral direction due to the final time ε and part O(ε³) in Eq. (2.43). In limit case when each motion time duration $\epsilon \to 0$ the obtained final pose is exactly in the lateral direction.

$\begin{array}{l}\hfill {\mathit{q}}_4=\left[\begin{array}{l}0\\ -{\epsilon }^2\\ 0\\ \end{array}\right]\end{array}$

As seen from Example 2.2 the Lie brackets can be used to identify new motion directions that are not allowed by the basic distribution. This new motion directions can be achieved by a finite number of infinitely small motions in directions from the basic vector field’s distribution. A Lie bracket is operating over two vector fields i(q) and j(q), and the result is a new vector field [i, j] obtained as follows:

$\begin{array}{l}\hfill [\mathit{i},\mathit{j}]=\frac{\partial \mathit{j}}{\partial \mathit{q}}\mathit{i}-\frac{\partial \mathit{i}}{\partial \mathit{q}}\mathit{j}\end{array}$

where

$\begin{array}{ll}\hfill \frac{\partial \mathit{i}}{\partial \mathit{q}}& =\left[\begin{array}{llll}\frac{\partial i_1}{\partial {\mathit{q}}_1}& \frac{\partial i_1}{\partial {\mathit{q}}_2}& \cdots & \frac{\partial i_1}{\partial {\mathit{q}}_n}\\ \frac{\partial i_2}{\partial {\mathit{q}}_1}& \frac{\partial i_2}{\partial {\mathit{q}}_2}& \cdots & \frac{\partial i_2}{\partial {\mathit{q}}_n}\\ ⋮& ⋮& ⋮\\ \frac{\partial i_n}{\partial {\mathit{q}}_1}& \frac{\partial i_n}{\partial {\mathit{q}}_2}& \cdots & \frac{\partial i_n}{\partial {\mathit{q}}_n}\\ \end{array}\right]\hfill \\ \hfill \frac{\partial \mathit{j}}{\partial \mathit{q}}& =\left[\begin{array}{llll}\frac{\partial j_1}{\partial {\mathit{q}}_1}& \frac{\partial j_1}{\partial {\mathit{q}}_2}& \cdots & \frac{\partial j_1}{\partial {\mathit{q}}_n}\\ \frac{\partial j_2}{\partial {\mathit{q}}_1}& \frac{\partial j_2}{\partial {\mathit{q}}_2}& \cdots & \frac{\partial j_2}{\partial {\mathit{q}}_n}\\ ⋮& ⋮& ⋮\\ \frac{\partial j_n}{\partial {\mathit{q}}_1}& \frac{\partial j_n}{\partial {\mathit{q}}_2}& \cdots & \frac{\partial j_n}{\partial {\mathit{q}}_n}\\ \end{array}\right]\hfill \end{array}$

Distribution is involutive if new linearly independent vector fields cannot be obtained using Lie brackets over the existing vector fields in the distribution. If distribution is involutive then it is closed under Lie brackets [6]. Then the system is completely holonomic with no motion constraints or only with holonomic constraints. This statement is explained by the Frobenious theorem [7]: If basic distribution is involutive then the system is holonomic and all existing constraints are integrable. Let us suppose that from m constraints, k constraints are holonomic and (m − k) nonholonomic. According to the Frobenious theorem there are three possibilities for k [8]:

• k = m, which means that the dimension of involutive distribution is (n − m), which also equals to the dimension of the basic distribution. Distribution dimension equals the number of linearly independent vector fields.

• 0 < k < m, k constraints are integrable, so k generalized coordinates can be eliminated from the system description. The dimension of involutive distribution is n − k.

• k = 0, the dimension of involutive distribution is n, and all constraints are nonholonomic.

Example 2.4

Determine motion constraints and directions of possible motion for a car-like vehicle without a (or with a locked) steering mechanism (Fig. 2.16). Find out the number and the type of constraints and the system’s DOF.

Solution

The vehicle cannot move in the lateral direction and it cannot rotate (its orientation φ is fixed); therefore constraint motion directions are

$\begin{array}{l}\hfill {\mathit{a}}_1(\mathit{q})=\left[\begin{array}{l}sin\phi \\ -cos\phi \\ 0\\ \end{array}\right]{\mathit{a}}_2(\mathit{q})=\left[\begin{array}{l}0\\ 0\\ 1\\ \end{array}\right]\end{array}$

and the motion constraints are

$\begin{array}{ll}\hfill {\mathit{a}}_1{(\mathit{q})}^T\dot{(}\mathit{q})=& \dot{x}sin\phi -\dot{y}cos\phi =0\hfill \\ \hfill {\mathit{a}}_2{(\mathit{q})}^T\dot{(}\mathit{q})=& \dot{\phi }=0\hfill \end{array}$

Both constraints are integrable, which can simply be seen by calculating their integral:

$\begin{array}{l}\hfill \begin{array}{c}\phi ={\phi }_0\\ (x-x_0)sin\phi -(y-y_0)cos\phi =0\end{array}\end{array}$

The vehicle can only move in the direction of the vector field:

$\begin{array}{l}\hfill {\mathit{s}}_1=\left[\begin{array}{l}cos\phi \\ sin\phi \\ 0\\ \end{array}\right]\end{array}$

Because both constraints are holonomic the distribution of the vector field s₁ is involutive, meaning that the system is completely holonomic and the system has 1 DOF. Using Lie brackets it is not possible to generate new vector fields as we have only one vector field that defines possible motion.