Motion planning for control-affine systems satisfying low-order controllability conditions

ABSTRACT This paper is devoted to the motion planning problem for control-affine systems by using trigonometric polynomials as control functions. The class of systems under consideration satisfies the controllability rank condition with the Lie brackets up to the second order. The approach proposed here allows to reduce a point-to-point control problem to solving a system of algebraic equations. The local solvability of that system is proved, and formulas for the parameters of control functions are presented. Our local and global control design schemes are illustrated by several examples.


Introduction
The motion planning problem for nonlinear systems has become an important research area over the last three decades due to its significant geometric features and applications in robotics. In spite of the number of studies, it still remains a challenging problem to construct control laws for general classes of systems, and the development of new approaches attracts considerable interest from both theoretical and applied points of view.
Let us briefly overview some related results in this area with a special emphasis on nonholonomic systems. Brockett (1981) proposed an optimal control law that steers first-order Lie bracket canonical systems. The construction of such optimal controls is also shown in the book by Bloch (2003). In Murray and Sastry (1990), an open-loop algorithm for steering first-and higher order chained-form systems using sinusoidal inputs has been proposed. A related method has been described in Sussmann and Liu (1991) for a more general class of driftless systems. In Liu (1997), a family of highly oscillatory high-amplitude inputs has been used for solving the problem of approximate tracking for a driftless control system. Highly oscillatory sinusoids are also applied in Gurvits and Li (1993) to compute time-periodic solutions for the nonholonomic motion planning problem with obstacle avoidance. A method for steering chained-form systems by piecewise-constant inputs is presented in Lafferriere and Sussmann (1991). Such type of controllers are used for the case of nilpotent systems as well as for the approximate steering problem of general CONTACT Alexander Zuyev zuyev@mpi-magdeburg.mpg.de nonholonomic systems. In Chumachenko and Zuyev (2009), the steering problem is solved for several examples of nonholonomic systems with piecewise-constant controls. Sinusoidal and polynomial inputs that steer a three-input system in two-chained form are constructed in Bushnell, Tilbury, and Sastry (1995). A globally convergent steering algorithm, based on nilpotent approximations, is proposed in Chitour, Jean, and Long (2013) and developed in the monograph by Jean (2014). The concept of interpolation entropy is introduced in Gauthier, Jakubczyk, and Zakalyukin (2010) to measure the asymptotics of the minimum length of admissible curves connecting the endpoints for the motion planning problem. In particular, it is shown that the entropy of a motion planning problem is equivalent to that of its nilpotent approximation. Estimates of the entropy and the metric complexity are obtained for generic motion planning problems by constructing their nilpotent approximations in Boizot and Gauthier (2013). A Lie algebraic method for motion planning exploiting the generalised Campbell-Baker-Hausdorff-Dynkin formula is described in Duleba, Khefifi, and Karcz-Duleba (2012).
To the best of our knowledge, only partial results are available for the control design of control-affine systems with drift. In Godhavn, Balluchi, Crawford, and Sastry (1999), motion planning algorithms with band-bang controls are presented for a class of Lagrangian systems with a cyclic coordinate. Another time-state controller for such type of systems is developed in Kiyota and Sampeio (1998). In Bloch and Reyhanoglu (1990), open-loop controls are obtained for a smalltime locally controllable (STLC) system describing the motion of a knife edge on a flat surface. The paper by Matsuno and Saito (2000) is devoted to the study of a class of control-affine systems with three states and two inputs. To produce a control law, the authors use a special chained-form transformation. The steering problem is considered in Basto-Gonçalves (1999) for control-affine systems under second-order STLC conditions. A discontinuous control law is developed in ur Rehman (2005) to steer a class of control-affine systems with zero drift at the origin. In Michalska and Torres-Torriti (2003), an approach for solving the stabilisation problem by a time-varying feedback law is proposed with the use of sampling strategy and nilpotent approximations of control-affine systems. The time-varying feedback law is constructed there by a concatenation of piecewise constant controllers. The parameters of such piecewise constant controllers are obtained from solving the 'satisficing problem' . An important step in this control design scheme requires the knowledge of solutions to the control-affine system with these parameters. Sufficient Lie algebraic conditions for the stabilisability of control-affine systems have been proposed in Tsinias and Theodosis (2015) by using sampled-data feedback laws and infinite partitions of the time interval.
In this paper, we consider a class of control-affine systems whose vector fields together with their first-and second-order Lie brackets satisfy Hörmander's condition. To solve a point-to-point control problem, we use a Volterra series development for solutions of the system with time-varying trigonometric inputs. The main contribution of this work concerns the construction of steering controls in Sections 3 and 4. This construction allows to compute the parameters of control functions in terms of solutions to auxiliary algebraic equations (Theorems 3.1 and 4.1). To the best of our knowledge, no solvability results have been available for this class of problems. Local solvability results (Theorems 3.2 and 4.2) are proved by exploiting the degree theory, and solutions to the approximate path-following problem are presented in Theorems 2.1 and 4.3. In Section 6, the results obtained are applied to solving the motion planning problem for several mechanical examples. Some technical details are presented in the Appendices.

Problem statement and approximation theorem
Consider a control-affine systeṁ where x = (x 1 , … , x n ) * is the state vector, u = (u 1 , . . . , u m ) * ∈ R m is the control, and ' * ' denotes the transpose. All vector fields f i : D → R n are assumed to be of class C 3 in a domain D.
For x 0 ࢠ D and an admissible control u : [0, τ ] → R m , we denote by x(t; x 0 , u) ࢠ D the solution of system (1) with initial data x| t = 0 = x 0 and control u = u(t), 0 ࣘ t ࣘ τ . We also use the notation B ε (x) = {y ∈ R n | x − y < ε} for an ε-neighbourhood of a point x ∈ R n , ρ(x, γ ) = inf y ࢠ γ x − y , and B ϵ (γ ) = Þ y ࢠ γ B ϵ (y) for an ε-neighbourhood of a set γ ⊆ R n . Here, · is the Euclidean norm on R n . To study the local steering problem, we introduce the class K whose elements are continuous strictly increasing functions θ : R + → R + such that θ(0) = 0, R + = [0, +∞). Problem 2.1 (Local approximate steering problem): For a given x α ࢠ D, ϵ 0 > 0, and with a constant r < 1 and a function θ ∈ K.
It is clear that if system (1) is locally controllable at a point x α ࢠ D, then, for small enough ϵ 0 > 0 and any x ω ∈ B 0 (x α ), there exists a control u x α x ω ∈ L ∞ [0, τ ] such that condition (2) holds with r = 0. However, in this paper, we treat the construction of controllers in Problem 2.1 as an algorithm that computes a smooth function u x α x ω (t ) in terms of solutions to certain algebraic equations whose coefficients depend on the vector fields f 0 (x), f 1 (x),..., f m (x), and, possibly, their Lie brackets at a point x = x α . We will also extend such an algorithm in order to follow a given curve γ in the state space D.
Problem 2.2 (Approximate path-following problem): For a given curve γ ࣪D with the endpoints x 0 and x T , and a given ε > 0, the goal is to construct a piecewisesmooth control u : For solving this problem, we use a partition π of γ with a finite number of points x j ࢠ γ , j = 0, 1, … , N: where '≺' denotes the natural order on γ . We assume for the moment that there are η > 0 and ϵ 0 > 0 such that Problem 2.1 is solvable for each x α ࢠ B η (γ ) and x ω ∈ B 0 (x α ) by a family of controls {u x α x ω (·)}, and that the mesh of π, defined as (π ) = max 1≤ j≤N x j − x j−1 , is small enough. Under these assumptions, we introduce the following definition.
As we will show in the proof of Theorem 2.1, the above construction is well defined if (π) is small enough.
Theorem 2.1: Let γ ࣪D be a curve with the endpoints x 0 and x T , and let positive numbers η, ϵ 0 be such that Problem 2.1 is solvable for each x α ࢠ B η (γ )࣪D and x ω ∈ B 0 (x α ) by a family of controls {u x α x ω (·)}. Assume, moreover, that the constant r ࢠ (0, 1) and the function θ ∈ K in formulas (2) and (3) may be chosen independently of x α ࢠ B η (γ ).
To complete the proof, we consider an arbitrary t ࢠ [t j , t j + 1 ] (0 ࣘ j ࣘ N − 1), and use the triangle inequality together with inequalities (3), (4) and (9): Thus, to prove that ρ(x(t), γ ) < ε, it suffices to show that Ifε = sup s∈R + θ (s) ≤ ε − ε 1 , then inequality (11) is satisfied with any¯ > 0. Otherwise, as the function θ ∈ K is strictly increasing on R + , inequality (11) is equivalent to¯ ≤ θ −1 (ε − ε 1 ) − ε 1 . The above inequality is satisfied, provided that condition (6) holds and¯ > 0 is given by formula (7). As 0 ࣘ j ࣘ N − 1 and t ࢠ [t j , t j + 1 ] may be taken arbitrarily, we have proved that Remark 2.1: The proof of Theorem 2.1 remains valid for general systems of the formẋ = f (x, u), x ࢠ D, u ࢠ U, as the main idea is just based on the group property: the translation of a trajectory is a trajectory for time-invariant control systems.
As we see, Theorem 2.1 justifies the possibility of reducing the approximate steering problem to successive concatenations of local controllers. Such local controllers will be constructed in this paper by exploiting the representation of solutions of system (1) by the Volterra series. Namely, if u(t) (0 ࣘ t ࣘ τ ) is an admissible control for system (1), then the corresponding solution x(t; x 0 , u) may be approximated by the Volterra series as follows (Lamnabhi-Lagarrigue, 1996;Nijmeijer & van der Schaft, 1990): where we introduce an artificial control u 0 ࣕ 1 for convenience of notation, ∂ f i (x) ∂x is the Jacobian matrix, and R(t; x 0 , u) is the remainder.
In Sections 3 and 4, we will present solutions to the local steering problem within the class of trigonometric polynomials as control inputs. Then, we will show how such controls can be used for solving the path-following problem in Section 4.

Controllability conditions with the first-order Lie brackets
For the local steering problem, our goal is to propose a control algorithm that steers system (1) from a given initial point x α ࢠ D to a small neighbourhood of a target point x ω ࢠ D at some time τ > 0. In order to solve this problem explicitly, we assume that there are sets of indices S 0 ࣮{1, 2, … , m}, S 1 ࣮{1, 2, … , m}, and S 2 ࣮{1, 2, … , m} 2 such that |S 0 | + |S 1 | + |S 2 | = n. Without loss of generality, we assume that the elements of S 2 are ordered such that i < j for each pair (i, j) ࢠ S 2 .
We consider the following family of controls: where k = 1, 2, … , m, v i , a i , a ij are real coefficients, K i and K ij are nonzero integers, and δ ki is the Kronecker delta.
For given x α , x ω ࢠ D and τ > 0, we will define the vector of coefficients (14) by using the following system of algebraic equations: with To formulate the basic result concerning the local steering problem, we need a non-resonance assumption concerning integer parameters K l and K ij . Assumption 3.1: For each l, q ࢠ S 1 and (i 1 , j 1 ) ࢠ S 2 , (i 2 , j 2 ) ࢠ S 2 such that l ࣔ q and (i 1 , j 1 ) ࣔ (i 2 , j 2 ), the following inequalities hold: Theorem 3.1: Assume that, for x α , x ω ࢠ D and positive numbers τ , ε, ε 1 , the vectors a ∈ R n and K ∈ (Z \ {0}) |S 1 |+|S 2 | satisfy the system of algebraic equations (15) and Assumption 3.1, and that the following conditions hold: Then, Here,B ε (x α ) stands for the closure of B ε (x α ), and The proof of Theorem 3.1 is given in Section 5.
Remark 3.1: By using the Taylor expansion, we conclude that condition (18) is equivalent to for small values ofŪ given by formula (20).
To study the solvability of algebraic equations (15), we introduce new variables and denote column vectors As each ϰ ij is the square root of a positive integer in (21), we will use the notation k}. We also introduce the n × n-matrix whose columns are formed by the vector fields from the rank condition (13) evaluated at x = x α . Then, we exploit formulas (16) and (21) to rewrite algebraic equations (15) in the following form: a ∈ R n is a solution of algebraic equation (23) with some ϰ ij > 0, then formula (21) implies that (15). We will prove the following local solvability result for system (23).
Theorem 3.2: Let the (S 1 , S 2 , S 3 )-rank condition (13) be satisfied at a point x α ࢠ D, and let (24) Then, for any small enough ϵ 0 > 0 and any x ω ∈ B 0 (x α ), the system of algebraic equations (23) has a solution ξ ∈ R n with some τ = O(ϵ 0 ) and (13) is satisfied, then the matrix A(x α ) given by (22) is nonsingular, and In order to prove the solvability of equations (23), we show that there exist positive numbers τ , ϵ 0 ,κ, ϵ w , ϵ a such that where Let us first consider the limiting caseκ → ∞. Then, inequality (26) takes the form We see that g τ (p, q) is increasing along l p and l q if If these conditions are satisfied, then formula (28) is reduced to In particular, condition (29) holds for With this choice of ϵ a and ϵ w , the inequalities where To satisfy condition (33), we observe that τ − γ τ 2 > τ 2 for τ ∈ 0, 1 2γ , γ > 0. This inequality implies that both conditions in (33) are satisfied for positive We note also that ϵ a and ϵ w are positive in formulas (31) if and only if Thus, by putting together the inequalities in (34) and (35), we conclude that, for any positive ϵ 0 such that 0 < min 1 2γ 2 , d 4γ 2 , dck 5 2(k 2 k 5 +2k 3 c) , inequality (28) holds with τ = 2 0 d and positive numbers ϵ a , ϵ w given by formula (31). It means also that property (26) Then, we choose the parameters κ i j ≥κ such that x ω (ξ ) denotes the right-hand side of Equation (23). Thus, the vector fields A(x α )ξ and (ξ ) = A(x α )ξ − x ω (ξ ) are homotopic on W, so the rotation of (ξ ) on W is equal to sign |A(x α )| ࣔ 0 (Krasnosel'skij & Zabrejko, 1984). Then, the principle of nonzero rotation implies that there exists a ξ ࢠ W such that (ξ ) = 0 (Zabrejko, 1997, Theorem 1), which completes the proof.

Second-order rank condition: nonholonomic systems
In this section, we consider a driftless control systeṁ Although the solvability of motion planning problems for nonholonomic systems has been already established under rather general controllability assumptions (Jean, 2014;Liu, 1997), our aim is to propose an explicit control design scheme and perform all necessary computations analytically. For this purpose, we restrict our analysis to a class of bracket-generating systems of step 3. Let S 2 ࣮{1, 2, … , m} 2 and S 3 ࣮{1, 2, … , m} 3 be subsets of indices such that |S 2 | + |S 3 | = n − m. Without loss of generality, we assume that the elements of sets S 2 and S 3 are ordered as j 1 < j 2 for all (j 1 , j 2 ) ࢠ S 2 , and l 1 < l 2 whenever (l 1 , l 2 , l 3 ) ࢠ S 3 .
In order to solve Problem 2.1, we apply the following family of control functions: where a k , a ij , a ijl are real coefficients, K ij , K 1ijl , K 2ijl are nonzero integer parameters. To define the vector of coefficients a=( a k | k∈{1,...,m} , a i j (i, j)∈S 2 , a i jl (i, j,l)∈S 3 ) * ∈R n , (39), we introduce the following system of algebraic equations: where the expression for is given in Appendix 2. We also need an extra non-resonance assumption on the frequencies of the sine and cosine functions, so that there are no low-order resonances among the frequency multipliers K ij , K 1ijl , K 2ijl , and K 1ijl ± K 2ijl .
Our basic result concerning solutions of the local steering problem for nonholonomic case is as follows.
Theorem 4.1: Assume that, for x α , x ω ࢠ D and positive numbers τ , ε, ε 1 , the vectors a ∈ R n and K ∈ (Z \ {0}) |S 2 |+2|S 3 | satisfy the system of algebraic equations (40) and Assumption 4.1, and that the following conditions hold: Then The proof of this result is given in Section 5. (42) is reduced to the following one:

Remark 4.1: For small values ofŪ , condition
A crucial assumption of Theorem 4.1 is that the coefficients of control (39) satisfy the system of algebraic equations (40). To prove the solvability of system (40), we introduce new variables In new variables, we write system (40) in the following form: where˜ (ã, x α ) does not contain terms of order less than 4/3 with respect toã (see Appendix 2). We assume that the (S 2 , S 3 )-rank condition is satisfied, therefore, the matrix is nonsingular. Then, we define the integers K + i j and K + 1i jl , K + 2i jl according to Assumption 4.1. Thus, ifã is a solution of system (46) for given x α , x ω ࢠ D, then the components of a solution of system (40) are where sign(ã i j ) = 1 ifã i j ≥ 0 and sign(ã i j ) = −1 otherwise. So, the solvability problem for system (40) is reduced to the study of system (46). The formula for (ã, x) in Appendix 2 implies that there exists a function C(x) > 0, which is continuous in D, such that We derive the following corollary of Theorem 4.1 for solving Problem 2.1.

Theorem 4.2:
Assume that the rank condition (38) holds at x = x α ࢠ D and that inequality (41) is satisfied inB ε (x α ) for some ε > 0. Then, for any r ࢠ (0, 1) and τ > 0, there exist ϵ 0 > 0 and θ ∈ K such that: (1) for any x ω ∈ B 0 (x α ), there exists a solution a ∈ R n of algebraic system (40) with some K ∈ (Z \ {0}) |S 2 |+2|S 3 | that satisfy Assumption 4.1; (2) if u(t) is the control given by formula (39) with the above a ∈ R n and K ∈ (Z \ {0}) |S 2 |+2|S 3 | , then, Proof: Let x α ࢠ D, ε > 0, r ࢠ (0, 1), and τ > 0 be given. To prove assertion (1), we note that solutions of algebraic systems (40) and (46) are related by transformation (48). We choose a vector K + ∈ N |S 2 |+2|S 3 | in such a way that Assumption 4.1 is satisfied. Then, we rewrite system (46) as In the trivial case x ω = x α , it is easy to see thatã = 0 ∈ R n is a root of algebraic equation (46). If x α − x ω > 0 is small enough, we will use the principle of nonzero rotation to prove that the equation ( We estimate the left-hand side of inequality (53) by using estimate (50) and assuming that d ࣘ 1: Thus, inequality (53) follows from the conditions where We see that the function μ x α (d) is positive and increasing on d ࢠ (0, d max ], where As μ x α (d) is strictly concave on R + and μ x α (0) = 0 , condition (54) is satisfied with Thus, we conclude that if then, condition (53) holds on the sphere S d of radius d given by formula (57). Thus, the maps (ã) and (ã) = a are homotopic on the sphere S d , and the rotation of (ã) is equal to 1. Applying the principle of nonzero rotation, we conclude that there exists anã ∈ B d (0) such that (ã) = 0 (see, e.g. Krasnosel'skij & Zabrejko, 1984). Then, we define the vectors a∈R n and K ∈ (Z \ {0}) |S 2 |+2|S 3 | by formulas (48) and (49) and observe that the system of algebraic equations (40) and Assumption 4.1 are satisfied. This completes the proof of assertion (1).
We will show below that the construction of local controllers in Theorem 4.2 can be used to satisfy the conditions of Theorem 2.1 for solving the approximate pathfollowing problem.

Theorem 4.3:
Let γ ࣪D be a curve with the endpoints x 0 and x T , and let the rank condition (38) be satisfied at each x ࢠ γ . Then, for any τ > 0 and ε > ε 1 > 0, there exists a¯ > 0 such that, for any partition π : Here, the control u π (t) is constructed as in Definition 2.1 by using the concatenation of local controllers u(t ) = u x α x ω (t ) of form (39) whose coefficients are defined by the system of algebraic equations (40).
Proof: As the rank condition (38) are finite by the Weierstrass theorem. We see that the conditions of Theorem 4.2 are satisfied for each x α ࢠ with the above choice of M 1 , M 2 , and M 3 . Let us now fix arbitrary r ࢠ (0, 1), τ > 0, and show that the number ϵ 0 > 0 and function θ ∈ K in Theorem 4.2 may be chosen independently of x α ࢠ . Since all the vector fields appearing in the rank condition (38) are continuous on the compact ࣪D, there exists a vector K + ∈ N |S 2 |+2|S 3 | satisfying Assumption 4.1 such that the matrix F(x α ) is nonsingular for each x α ࢠ . As in the proof of Theorem 4.2, we fix such K + ∈ N |S 2 |+2|S 3 | and introduce the func- and μ(d) > 0 is strictly increasing on d ∈ (0,d max ], d max = min 1, 3 4c 1 c 2 3 . Following the proof of Theorem 4.2 with the use of inequality (68), we conclude that its assertions (1) and (2) remain true for each  (60) and (67), respectively, and Thus, expression (69) defines the constant ϵ 0 > 0 for Theorem 4.2 independently of x α ࢠ . It remains to verify that there exists a θ ∈ K such that the estimate holds for each x α ࢠ and θ x α (s) given by formula (64). Indeed, straightforward computations with the use of inequality (68) show that the function Thus, we have shown that formulas (69) and (72) define the constant ϵ 0 > 0 and function θ ∈ K for Theorem 4.2 independently of x α ࢠ . Now, the assertion of Theorem 4.3 follows from Theorem 2.1.
In Section 6, we demonstrate the approach of Theorem 4.3 with several examples, where the system of algebraic equations (40) will be solved numerically.

Auxiliary results and proofs
To prove Theorem 3.1, we rewrite the Volterra series (12) by using the first-order Lie brackets as follows: where R 2 (t) is the sum of the last two terms of formula (12). We need two auxiliary lemmas, whose proofs can be found in Zuyev (2016).
Lemma 5.1: LetD ⊂ R n be a closed convex domain, and let x(t ) ∈D, 0 ࣘ t ࣘ τ , be the solution of system (1) corresponding to initial value inD with some positive constants M 1 and M 2 , then the remainder R 2 (τ ) of the Volterra expansion (73) satisfies the following estimate: Lemma 5.2: Let x(t ) ∈D ⊂ R n , 0 ࣘ t ࣘ τ , be a solution of system (37) with a control u ࢠ C [0, τ ], and let = 1, 2, . . . , m. Then, (14) into formula (73) with x 0 = x α ࢠ D and computing the integrals, we obtain

Proof of Theorem 3.1: By substituting controls
where the terms V 20 and V 21 are given by formulas (16) provided that Assumption 3.1 holds. For given x α , x ω ࢠ D and τ > 0, we assume that the vector a = (v i ) i∈S 0 , (a i ) i∈S 1 , (a i j ) (i, j)∈S 2 * ∈ R n satisfies the system of algebraic equations (15) and K ∈ (Z \ {0}) |S 1 |+|S 2 | satisfies Assumption 3.1. Then, formulas (15) and (77) imply is the solution of system (1) with the control u = u(t) of form (14). Thus, it suffices to prove that and We estimate the sum of components |u i (t)| in formula (14) as follows: Hence, is given in (20). As the right-hand side of inequality (75) is strictly increasing with respect to U ∈ R + and U τ ≤Ū , inequality (78) follows from condition (18) because of Lemma 5.1 withD =B ε (x α ). To show that inequality (79) holds, we apply a modification of estimate (76) for system (1). Indeed, the assertion of Lemma 5.2 for system (1) can be formulated as follows: whereŪ , M 0 , and M 1 are defined in (17) and (20). Now, inequality (79) follows from conditions (19) and (80).
In order to prove Theorem 4.1, we rewrite formula (12) by using the Lie brackets as follows: The proof of this fact is presented in Appendix 1 together with the expression for G(t), and the remainder R(t) is estimated by the following lemma.

Proof: Let us denote by R (N+1)
i (x) the remainder term for the N-th order Taylor expansion of f i (x) at a point x 0 ∈ D. If f i (x) is of class C N + 1 in a convex domainD, then R (N+1) (x) may be represented in the Lagrange form of the remainder as follows: To prove the assertion of Lemma 5.3, we use the integral representation of system (37) with initial conditions x(0) = x 0 and the Taylor expansion for f ik (x): where the gradient ∂ f ik (x) ∂x is treated as a row vector. After several transformation, expression (85) takes the form (12) with By estimating the absolute value of R k (t) term-by-term in (86) with the use of (84), we get whereM 2 = 1 2 sup x∈D |α|=2 The norm of x(t) = x(t) − x 0 is estimated by Lemma 5.2 as follows: As the function ψ(β) = e β − 1 is convex, it follows from (89) that Component-wise estimates (87) together with inequalities (88) and (90), and U 2 t 2 ≤ŪUt, 0 ≤ Ut ≤Ū imply estimate (83) for the Euclidean norm of R(t).
Proof of Theorem 4.1: By substituting the control u = u(t) of form (39) into the Volterra series (81) with provided that Assumption 4.1 is satisfied (the explicit formula for is in Appendix 2). It is easy to see that the system of algebraic equations (40) is equivalent to the following condition in terms of representation (91): Therefore, if the vectors a ∈ R n and K ∈ (Z \ {0}) |S 2 |+2|S 3 | satisfy the system of algebraic equations (40) and Assumption 4.1, then it remains to show that because of Lemma 5.2 withD =B ε (x α ), where U = max 1≤i≤m m i=1 |u i (t )| ≤Ū /τ, the constants M i are given in formulas (41) and (44). To complete the proof, we conclude that conditions (92) follow from Lemma 5.3 and inequalities (42) and (43).
The value of x(τ ) − x ω from Figure 2(f) can be used to evaluate the relative accuracy of our local steering algorithm:r = x(τ ) − x ω / x α − x ω ≈ 0.027 < 1. Note that a theoretical upper bound forr is given by the r constant in (2) (Problem 2.1 formulation). This constant can be estimated from Theorem 4.1 as r = φ(Ū )/ x α − x ω , where the computation of φ(Ū ) by formula (42) is based on the coefficients a of the control (94) and the upper bounds of the derivatives of f i (x). Similarly, the maximal overshoot is estimated by inequality (3): , where the right-hand side can be estimated as θ (s) = θ x α (s) by formula (64) from the proof of Theorem 4.2.
Note that system (99) is a modification of the equations considered in Nalamura and Savant (1991) for the case when the angular velocity component along the x 3 axis is not controlled (u 0 = const). Therefore, our controls are the translational velocity u 1 = v along the Ox 1 axis and two angular velocity components: u 2 = ω 1 and u 3 = ω 2 .
The above parameters are obtained by solving the system of algebraic equation (15) with x α = x((j − 1)τ ), x ω = x j , and the integer parameters being chosen as K 1 = 3, K 13 = 1, K 23 = −2 (these parameters clearly satisfy Assumption 3.1). We see in Figure 4 that the controller proposed is able to solve the approximate path-following problem for system (99) with the accuracy x(T) − x T < ε 1 0.002 at the final time T = 0.4.

Conclusion
In this paper, we have proposed an explicit reduction of the motion planning problem to systems of algebraic equations for classes of bracket-generating systems of steps 2 and 3. To the best of our knowledge, no general results concerning the solvability of such algebraic systems of an arbitrary dimension have been published so far. On the one hand, it has been already proved in Liu (1997) that any trajectory of the Lie bracket extension can be approximated by trajectories of the original system with highly oscillatory inputs. On the other hand, we do not use any sequence of trigonometric polynomials with unbounded amplitudes and frequencies here. It should be also emphasised that our construction provides explicit formulas for controls and does not use any specific changes of coordinates (e.g. canonical coordinates corresponding to the P. Hall basis). Thus, our solvability result provides a novel contribution towards the justification of the use of trigonometric controls for local and global steering problems. Note that the proofs of Theorems 3.2 and 4.2 are based on the degree theory, as the standard implicit function theorem is not applicable (the nonlinear part of the corresponding vector function is not differentiable atã = 0).