Adaptive Motion/force Control of Nonholonomic Mechanical Systems with Affine Constraints *

The motion/force tracking control of nonholonomic mechanical systems with affine constraints is investigated in this paper. In the procedure of control design, by flexibly using the algebra processing technique, constraint forces are successfully canceled in the dynamic equations, and then an integral feedback compensation strategy and an adaptive scheme are applied to identify the dynamic uncertainty. It is shown that the proposed controller ensures the position state of the closed-loop system asymptotically tracks the desired trajectory and the force tracking error has a controllable bound. Finally, a boat on a running river as a simulation example is given to show the effectiveness of the control scheme.


Introduction
Nonholonomic constraints arise in many mechanical systems when there is a rolling or sliding contact, such as wheeled mobile robots, n-trailer systems, space robots, underwater vehicles, multi-fingered robotic hands, and so on.Although great progress [1][2][3][4][5][6][7][8][9] has been made for nonholonomic systems during the last decades, controller design for these systems still has a challenge to control engineers.
It is worth pointing out that most existing results [10][11][12][13][14][15] aimed at the classic nonholonomic linear constraints (i.e.J(q) q = 0).In fact, there is another large class of constraints, which are affine in velocities, called affine constraints [16][17][18][19] (i.e.J(q) q = A(q)), such as a boat on a running river with the varying stream, ball on rotating table with invariable angular velocity, underactuated mechanical arm, etc. Kai defined rheonomous affine constraints in [16] and explained a geometric representation method for them, and derived a necessary and sufficient condition for complete nonholonomicity of the rheonomous affine constraints.In [17], Kai derived very good results about nonholonomic dynamic systems with affine constraints.To be specific, the local accessibility and local controllability were developed based on both Sussmann's theorem and linear approximation approaches, and a necessary and sufficient condition for complete nonholonomicity of the a rheonomous affine constraints were presented at last.
The tracking problem for mechanical systems, as a much more interesting issue in practice, is to make the entire state of the closed-loop system track to a given desired trajectory.For example, [20] presented an adaptive robust control strategies for a class of mechanical systems with both holonomic and nonholonomic constraints.Based on physical properties, adaptive robust motion/force control for wheeled inverted pendulums is investigated in [21].For mobile manipulators under both holonomic and nonholonomic constraints, [22] and [23] proposed state-feedback control strategies by introducing an appropriate state transformation, and an adaptive robust output-feedback force/motion control strategies, respectively, and so on.However, it should be noted that the aforementioned works were reported on nonholonomic systems with classical linear constraints, and to date, no solutions have been done on tracking control of nonholonomic systems subjected to affine constraints.Hence, researching the tracking problem for such nonholonomic mechanical systems is an innovatory and significative work.Xian presented a new continuous control mechanism that compensated for uncertainty in a class of high-order, multiple-input multiple-output nonlinear systems in [24].Based on this control strategy, Makkar considered modeling and compensation for parameterizable friction effects for a class of mechanical systems [25].However, the upper bound of the uncertainties must be known as a prerequisite.Base on Xian's compensatory scheme, this paper investigates the tracking control problem for a class of uncertainty nonholonomic mechanical systems.At the same time, the constraint that the upper bound of the uncertainties is known is relaxed by constructing an adaptive update law.To achieve the tracking objective, by flexibly using the Algebra processing technique, we firstly triumphantly reduce the number of state variables, which provide a motion complying with affine constraints, and an integral feedback compensation term is used to identify the dynamic uncertainties.The main contributions of the paper are briefly characterized by the following specific features: (i) We establish the dynamical model of the nonholonomic control systems with affine constraints.Since affine constraints are introduced to mechanical systems, it is difficult to find linearly independent vector fields to cancel the constraint forces J(q)λ in dynamic equation.Hence, how to deal with them is the main innovation of this paper.(ii) Based on the asymptotic tracking idea for uncertain multi-input nonlinear systems, the related adaptive theory and the compensatory strategy for the uncertainties, an adaptive tracking controller is designed such that the trajectory tracking error asymptotically tends to zero and the force tracking error is bounded with a controllable bound.
(iii) As a practical application, a boat on a running river with varying stream is presented to illustrate the reasonability of the assumptions and the effectiveness of the control strategy.
Notations.x denotes the Euclidean norm of x; C i denotes the set of all functions with continuous ith partial derivative on R n ; sgn(•) denotes the standard signum function; a continuous function h : R + → R + is said to be a K function if it is strictly increasing and vanish at zero.For simplicity, sometimes the arguments of functions are dropped.
2 System description and control design

Dynamics model
According to Euler-Lagrange formulation, equations of nonholonomic mechanical systems are described by where q = [q 1 , . . ., q n ] T is the generalized coordinates, and q, q ∈ R n represent the generalized velocity vector, acceleration vector, respectively; M (q) ∈ R n×n is the inertia matrix; V (q, q) q ∈ R n presents the vector of centripetal, Coriolis forces; G(q) ∈ R n represents the vector of gravitational forces; τ denotes the r-vector of generalized control inputs; B(q) ∈ R n×r is a known input transformation matrix (r < n) with full rank; f ∈ R n denotes the vector of constraint forces.Consider the situation where kinematic constraints are imposed, which represented by analytical relations between the generalized coordinates q and velocity vector q, it is can be described by where J(q) = [j 1 (q), . . ., j m (q)] ∈ R n×m is full of constraint matrix, A(q) = [a 1 (q), . . ., a m (q)] T ∈ R m is known.The constraint equation ( 2) is regarded as affine constraints.When it is imposed on the mechanical systems (1), the constraint forces are given by f = J(q)λ, where λ ∈ R m is a Lagrangian multiplier corresponding to m nonholonomic affine constraints.
Remark 1. (i) It's worth pointing out that the system studied in this paper is much more general than that in some existing literatures such as [10,14,[20][21][22][23], in which dynamic equations satisfy the classical linear constraints.In fact, by taking A(q) = 0, J T (q) q = A(q) can be transformed into linear constraints, whose tracking problems have been extensively studied during the last two decades.
(ii) With affine constraints introduced to mechanical systems, traditional methods [20][21][22][23] are hardly applied to such systems.Hence, how to deal with constraint forces is also a difference between this paper and the existing literatures.

Reduced dynamics and state transformation
This part mainly focuses on reducing the number of state variables, which provides motion complying with the affine constraints.
It is easy to find a full-rank matrix S ∈ R n×(n−m) satisfying Noticing that S(q) is full of rank, there must exist a full-rank matrix S 1 (q) ∈ R (n−m)×n satisfying S 1 (q)S(q) = I, where I is an identity matrix.If defining can be expressed concisely as For the sake of convenience, we define where η(q) ∈ R n satisfies J T (q)η(q) = A(q).One can deduce that E is a full of rank and satisfies J T (q), A(q) E(q) = 0.
From ( 4) and ( 5), we know that there exists an where In view of the relationship (6), one can obtain żn−m+1 = 1, and the generalized velocity vectors can be written as z corresponds to the internal state variable, and (q, z) is sufficient to describe the constrained motion.Equation (7) represents the kinematics of a nonholonomic mechanical system.Substituting ( 7) into (1), the dynamics of the mechanical system (1) with affine constraints (2) can be described clearly as where M (q) = M (q)S(q) and V (q, q) = M (q) Ṡ(q) + V (q, q)S(q), Ḡ(q, q) = M (q) η(q) + V (q, q)η(q) + G(q) can be seen as the generalized gravitational force vector.
Remark 2. The aforementioned transform methods differ from the traditional ones mentioned in existing papers [20][21][22].More specifically, when the affine constraints are imposed on the mechanical systems, it is difficult to find linearly independent vector fields to proceed a simple transformation for canceling the constraint forces in dynamic equations.Hence, we present a new transformation to achieve this goal.
Remark 3. The transformation consists of ( 3) and (7) can ensure system (8) still satisfy constraint equation ( 2), and possesses the practical physical meaning, this can also be confirmed by the practical example in Section 5.
Assumption 1. (See [26].)The matrix M 1 is symmetric, positive definite and satisfies where a is a known positive constant, ā(x) is a known positive function.
For practical mechanical systems, the Assumption 1 and 2 are reasonable.Next, we develop the following tracking error system, which will be used in the subsequent controller design and stability analysis: where e 1 ∈ R n−m , e λ ∈ R m .To achieve the desired control objective, the following filtered tracking errors [25,27], denoted by e 2 , ρ ∈ R n−m , are defined as where α 1 > 0, α 2 > 0 are designed constants.
In view of ( 9), ( 11) and ( 13), pre-multiplying M 0 on both sides of the second formula of (13), the following expression can be arrived at: Based on the expression ( 14), a control torque input is designed as follows: where the force term λ c is defined as λ c = λ d − k λ e λ , k λ is a constant of force control feedback gain, and µ(t) ∈ R n−m denotes a subsequently designed control term.Substituting ( 15) into ( 14), we can further get After pre-multiplying S T , noting S T (q)J(q) = 0 and S 1 (q)S(q) = I, the above equation becomes To facilitate the design of µ(t), differentiating (17) yields where Based on the method of compensation for uncertain dynamic [24], µ(t) is designed as follows: with the adaptive update law where design parameters k s , γ ∈ R are positive control gains, and Θ(t) ∈ R is the parameter estimation of Θ, which will be specified later.The second term in ( 19) is used to ensure that µ(0) = 0. µ(t) does not depend on the unmeasurable filtered tracking error term ρ, but its time derivative can be expressed as a function of ρ.Taking the time derivative of µ(t), one has Substituting ( 21) into ( 18) results in where Γ (z, ż, t) = Φ1 + Υ − (1/2) Ṁ1 ρ + e 2 ∈ R n−m .Now, defining Noting that Φ 1 is an C 2 vector function and z d , i = 0, . . ., 4, are all bounded, there must exist two unknown positive constants B 1 and B 2 such that Defining Γ (t) = Γ (t) − Γ d (t), the closed-loop error system ( 22) can be rewritten as Remark 4. It is worth highlighting that Θ(t) does not depend on the unmeasurable signal z(t).In fact, integrating both sides of formula (20)

Main results
Now, we are ready to present the following theorem, which summarizes the main results of the paper.
Theorem 1.Consider the nonholonomic mechanical system described by (1) and (2), subjects to Assumptions 1 and 2. Given a desired trajectory z d (t), which satisfies the constraint equation (2), using the control laws (15), ( 19) and (20), the following hold: (i) All the states of the closed-loop system are globally bounded.
(ii) The tracking error e 1 and ė1 converge to zero as t → ∞.
(iii) e λ is bounded for all t 0.
Proof.Let D ∈ R 3(n−m)+2 be a domain containing y(t) = 0, where y(t 1 , e T 2 , ρ T ] T , and Θ(t) = Θ − Θ(t) represents the parameter estimation error.The function P (t) ∈ R is defined as where the auxiliary function L(t) is defined as Selecting Θ = B 1 + (1/α 2 )B 2 + 1, by taking the same manipulations as Appendix A in [24], there is Hence, P (t) 0. Now, choose a continuously differentiable, positive definite and radially unbounded function Taking the time derivative of V along solutions of (10), noting the definition of Θ and substituting ( 11), ( 13) and ( 23) into it, we immediately get Since Γ (t) is continuously differentiable, with the the help of mean valve theorem, one can acquire the upper bound of Γ as follows [24]: where ϕ : R + → R + is an appropriate K function.By using the fact that 2e T 1 e 2 e 1 2 + e 2 2 , V can be simplified as where λ = min{2α 1 − 1, α 2 − 1, 1}, and α 1 , α 2 must be chosen to satisfy Completing the squares for the third term in (26), it follows that with this inequality in mind, the following expression can be obtained: Now, we define a compact set Inequality (27) shows V (t) V (0) in N 1 , hence, all the the signals e 1 , e 2 , ρ, Θ on the right-hand side of function (24) are bounded in N 1 .From the definition of e 1 , e 2 , ρ, Θ, we know ė1 = e 2 − α 1 e 1 , ė2 = ρ − α 2 e 2 , Θ = Θ − Θ, therefore, we can further get ė1 , ė2 , Θ ∈ L ∞ in N 1 .The assumption that z d , żd , zd are bounded can be used to conclude that z, ż, z ∈ L ∞ in N 1 .With M 1 , V 1 , G 1 are all known and bounded in N 1 .Thereby τ 1 , µ ∈ L ∞ in N 1 can be further obtained from (15) and (19).

Simulation
Consider a boat with payload on a running river [17,19] (see Fig. 1).The x-axis and y-axis denote the transverse direction and the downstream direction of the river, respectively.Here, we suppose the stream of the river only depends on transverse position x in the simulation.According to the motion of boat on the river, one can get the following kinematic equations: where C(x) denotes the stream of the river.After some simple calculations, the affine constraints can be obtained as follows: We assume that the traveling direction velocity and the angular velocity of the boat can be controlled.Hence, the control input τ = [τ 1 , τ 2 ] T is defined by the transformation matrix The standard forms are given as follows: where m is the mass of the boat and I is the inertia of the boat, m 0 denotes the unknown mass of the payload and I 0 denotes the unknown inertia of the payload.For the sake of simplicity, select m = 1, I = 1, C(q 2 ) = q 2 .One can choose It follows from the procedure of the aforementioned diffeomorphism transformation that q1 = ż1 sin q 3 + q 2 , q2 = ż1 cos q 3 , q3 = ż2 .
After imposing the constraint forces, the original system can be converted into the following form: where Π 1 = z1 sin z 2 + ż1 ż2 cos z 2 + ż1 cos z 2 , Π 2 = z1 cos z 2 − ż1 ż2 sin z 2 .For the given J(q), S(q) and η(q), the desired trajectory q d = [sin t−cos t, sin t, π/4] T satisfies kinematic constraint J T (q d ) qd = A(q d ) and diffeomorphism transform qd = S(q d ) żd + η(q d ) with z d = [ √ 2 sin t + 2, π/4] T .The control objective is to determine an adaptive feedback control so that the trajectory z follows z d , and λ is bounded.
Based on the previous design procedure, we get the actual controller www.mii.lt/NAFig. 2. The trajectories of e 1 (t).The results of the simulation are shown in Figs.2-5. Figure 2 shows the position tracking errors of z(t) − z d (t) converge to zero, Fig. 3 shows the velocity tracking errors of ż(t) − żd (t) converge to zero, Fig. 4 shows both state Θ(t) and the tracking error of e λ are bounded.It can be seen that the control inputs shown in Fig. 5 are bounded.At the last of the simulation, we should explain why the given signal z 2d is a constant.In fact, according to transformation (28), we know z 2 = θ, so the control torques ensure asymptotical tracking all the time in the unchanged yaw angle with the different velocity of flow.Hence, the practical simulation example confirms the validity of the proposed algorithm.

Conclusions
In this paper, the trajectory and force tracking problem is addressed for a class of uncertain nonholonomic mechanical systems.The controller guarantees that the configuration state of the system semi-global asymptotically tracks to the desired trajectory and the force tracking error is bounded with a controllable bound.A practical mechanical model is constructed to confirm the reasonability of assumptions and the effectiveness of the control scheme.

d
(t), i = 0, . . ., 4, exist and are bounded, a desired constraint force f d (t) or a desired multiplier λ d (t), determine a adaptive control law for system (1) such that:(i) All the states of the closed-loop system are globally bounded.(ii) The position and velocity tracking error z(t) − z d (t), ż(t) − żd (t) converge to zero as t → ∞, respectively.(iii) The tracking error of constraint force f − f d is bounded for all t 0.