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International Journal of Control Vol. 00, No. 00, Month 200x, 1–12
RESEARCH ARTICLE
Iterative Learning Control and the Singularity Robust Jacobian Inverse for Mobile Manipulators Krzysztof Tcho´ n Institute of Computer Engineering, Control and Robotics Wroclaw University of Technology ul. Janiszewskiego 11/17 50–372 Wroclaw, Poland email:
[email protected] (Received 00 Month 200x; final version received 00 Month 200x) The method of iterative learning control to large extent has been inspired by robotics research, focused on the control of stationary manipulators. In this paper we deal with the inverse kinematics problem for mobile manipulators, and show that a very basic singularity robust Jacobian inverse can be derived in a natural way within the framework of iterative learning control. To achieve this objective we have exploited the endogenous configuration space approach. The introduced Jacobian inverse defines the singularity robust Jacobian inverse kinematics algorithm for mobile manipulators. A Kantorovich-type estimate of the region of guaranteed convergence of the algorithm is derived. For two example kinematics this estimate has been computed efficiently.
Keywords: mobile manipulator, inverse kinematics, iterative learning control, Jacobian algorithm
1
Introduction
A mobile manipulator is a robotic system that consists of a mobile platform and a stationary (i.e. with fixed base) manipulator mounted on the platform. In this paper we study nonholonomic platforms equipped with holonomic on board manipulators. Specifically, using the endogenous configuration space approach (Tcho´ n and Muszy´ nski 2000, Tcho´ n and Jakubiak 2003), we address the inverse kinematics problem for mobile manipulators. The problem amounts to computing controlled velocities of the platform and joint positions of the on board manipulator guaranteeing that at given time instant the end effector of the manipulator will reach a prescribed position and orientation. Solving the inverse kinematics problem forms indispensable ingredient of robot motion planning algorithms (LaValle 2006). Most frequently, the inverse kinematics problem is solved by means of a Jacobian inverse kinematics algorithm. A number of such algorithms for mobile manipulators have been derived in (Tcho´ n and Jakubiak 2003) and (Tcho´ n et al. 2010). There are two obstacles impairing the performance of Jacobian algorithms: ill definiteness at kinematic singularities and local convergence. To prevent the former, a singularity robust Jacobian inverse kinematics algorithm for mobile manipulators has been devised (Tcho´ n and Jakubiak 2003), that generalizes its well known analog for stationary manipulators invented by Nakamura (1991). Addressing the latter obstacle, Tcho´ n and Malek (2009) have proved that in the continuous case the singularity robust inverse Jacobian algorithm is globally convergent. To our best knowledge, global convergence of other Jacobian inverse kinematics algorithms has been proved only for very particular classes of mobile manipulators (Malek 2009). Robustness against singularities and global convergence distinguish the singularity robust Jacobian inverse kinematics algorithm from among alternative algorithms. This paper presents a reconstruction ISSN: 0020-7179 print/ISSN 1366-5820 online c 200x Taylor & Francis
DOI: 10.1080/0020717YYxxxxxxxx http://www.informaworld.com
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of this algorithm within the framework of iterative learning control, and shows that the iterative learning control approach inspires new results in robot motion planning. Iterative learning control offers a collection of tools applicable to repeatedly operated systems, and allowing them to improve their response in subsequent trials. A robotic perspective has been accompanying the iterative learning control since its very beginning, to mention the fundamental paper by Arimoto et al. (1984), devoted to stationary manipulators. Example application of iterative learning control to mobile robots can be found in (Oriolo et al. 2000). Diverse theoretical and practical issues of iterative learning control have been addressed in (Owens et al. 1995, Longman 2000, 2003, Avrachenko and Longman 2003). A very recent account of iterative learning control techniques with focus on real-time implementation can be found in (Xu et al. 2009). This paper addresses the inverse kinematic problem for mobile manipulators, and solves this problem using the singularity robust Jacobian inverse kinematics, whose operation improves gradually from step to step until a solution of the problem is reached. By design, the algorithm operates in a Hilbert space of endogenous configurations. There are two main results of the paper. First, we have demonstrated that the singularity robust Jacobian inverse for mobile manipulators (so, a fortiori, for stationary manipulators and nonholonomic mobile platforms) can be derived in a natural way within the framework of iterative learning control. Actually, the singularity robust Jacobian inverse may be regarded as a sort of Tikhonov regularization of the Jacobian pseudo inverse operator (Avrachenko and Longman 2003). Second, we have provided an estimate for the region of convergence of this algorithm operating in sampled mode, depending on the sample rate and the regularization parameter. In agreement with what has been stated above, the convergence region grows up to infinity when the sampling rate approaches zero. For two example kinematics the estimate has been computed explicitly. The composition of this paper is the following. In section 2 we briefly summarize basic concepts referring to the endogenous configuration space approach. Section 3 contains a derivation of the singularity robust Jacobian inverse for mobile manipulators. Section 4 addresses the problem of convergence of the singularity robust Jacobian inverse kinematics algorithm. Section 5 contains computations. The paper is concluded with section 6.
2
Basic concepts
As mentioned in the Introduction, we shall consider mobile manipulators composed of a nonholonomic mobile platform and a holonomic on board manipulator. The kinematics of such mobile manipulators can be represented in the following form of a driftless control system with outputs, q˙ = G(q)u =
m P
gi (q)ui , y = k(q, x),
(1)
i=1
where q ∈ Rn denotes the platform posture, u ∈ Rm is the control vector, x ∈ Rp describes the joint position of the manipulator, and y ∈ Rr refers to end effector coordinates in the task space. The task space is assumed Euclidean. In most cases the the number of controls does not exceed the dimensionality of the task space, i.e. m ≤ r. Admissible control functions u(·) of the platform will be assumed Lebesgue square integrable on a time interval [0, T ]. These functions along with joint positions of the on board manipulator constitute the endogenous configuration space X = L2m [0, T ]× Rp . Endogenous configurations play the role of inputs to system (1). Given an endogenous configuration (u(·), x) ∈ X , we compute the platform trajectory q(t) = ϕq0 ,t (u(·)), the end effector trajectory y(t) = k(q(t), x), and define the kinematics Kq0 ,T : X → Rr of the mobile manipulator as Kq0 ,T (u(·), x) = y(T ) = k(ϕq0 ,T (u(·)), x).
(2)
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The kinematics determine reachable at T end effector positions and orientations of the mobile manipulator subject to the input (u(·), x), provided that the platform starts from the initial posture q0 . The map (2) is continuously differentiable, (Bismut 1984), Theorem 1.1. The endogenous configuration space constitutes a Hilbert space with inner product
< (u1 (·), x1 ), (u2 (·), x2 ) >RW =
ZT
uT1 (t)R(t)u2 (t)dt + xT1 W x2
(3)
0
defined by symmetric, positive definite matrices R(t) and W . The induced RW -norm of the endogenous configuration ||(u(·), x)||2RW
=
ZT
uT (t)R(t)u(t)dt + xT W x.
(4)
0
The next important concept is the Jacobian operator
ZT Jq0 ,T (u(·), x)(v(·), w) = DKq0 ,T (u(·), x)(v(·), w) = C(T, x) Φ(T, s)B(s)v(s) ds + D(T, x)w, 0
(5) obtained by differentiation of the kinematics at (u(·), x). The Jacobian may be identified with the output reachability map at T of the linear approximation to system (1) along a triple inputtrajectory (u(t), x, q(t)), ξ˙ = A(t)ξ + B(t)v,
(6)
η = C(t, x)ξ + D(t, x)w,
initialized at ξ0 = 0. The matrices appearing in (6) are computed in a standard way A(t) =
∂ (G(q(t))u(t)) , ∂q
B(t) = G(q(t)),
C(t, x) =
∂k(q(t), x) , ∂q
D(t, x) =
∂k(q(t), x) , ∂x
∂ while the matrix Φ(t, s) satisfies the evolution equation ∂t Φ(t, s) = A(t)Φ(t, s) with initial condition Φ(s, s) = In . The Jacobian (5) is a linear transformation of the endogenous configuration space into the task space. Its dual operator,
Jq∗0 ,T (u(·), x) : (Rr )∗ −→ X ∗ , acting between respective dual spaces, called the adjoint Jacobian of the mobile manipulator, is defined in the following way
Jq∗0 ,T (u(·), x)η (v(·), w) = hJq∗0 ,T (u(·), x)η, (v(·), w)iRW = η T Jq0 ,T (u(·), x)(v(·), w) = ZT η T C(T, x) Φ(T, t)B(t)v(t)dt + D(T, x)w = 0
h(R
−1
(·)B T (·)ΦT (T, ·)C T (T, x)η, W −1 D T (T, x)η), (v(·), w)iRW , (7)
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where η ∈ (Rr )∗ ∼ = X . In consequence, the adjoint Jacobian = Rr and (v(·), w) ∈ X ∗ ∼
Jq∗0 ,T (u(·), x)η (t) = R−1 (t)B T (t)ΦT (T, t)C T (T, x), W −1 D T (T, x) η.
(8)
Endogenous configuration (u(·), x) ∈ X is referred to as regular, when the Jacobian is an onto map, otherwise this configuration is singular. In may be shown that the configuration (u(·), x) is regular, provided that the dexterity matrix Dq0 ,T (u(·), x) = Jq0 ,T (u(·), x)Jq∗0 ,T (u(·), x) = D(T, x)W −1 D T (T, x)+ C(T, x)
ZT
Φ(T, s)B(s)R−1 (s)B T (s)ΦT (T, s)dsC T (T, x) (9)
0
of the mobile manipulator has rank r. Given a point yd in the task space of a mobile manipulator with kinematics (2), the inverse kinematics problem for the mobile manipulator consists in determining an endogenous configuration (u(·), x) ∈ X , such that the task space error e = Kq0 ,T (u(·), x) − yd
(10)
is equal to zero.
3
Singularity robust Jacobian inverse
The inverse kinematic problem can be solved in accordance with the basic iterative learning control scheme (Arimoto et al. 1984, Owens et al. 1995, Longman 2000), (uk+1 (·), xk+1 ) = (uk (·), xk ) − L(uk (·), xk )ek = (uk (·), xk ) + (∆uk (·), ∆xk ),
(11)
k = 0, 1, . . . , including a linear learning operator L, where ek = Kq0 ,T (uk (·), xk ) − yd denotes the error (10) at the step k. We shall assume that the increments ∆uk (·) = uk+1 (·) − uk (·), ∆xk = xk+1 − xk , of the endogenous configuration at the step k + 1 minimize the performance index I(∆uk (·), ∆xk ) = ||ek+1 ||2 + κ||(∆uk (·), ∆xk )||2RW ,
(12)
where the error norm is Euclidean, the RW -norm has been defined by (4), and κ > 0 denotes a positive number referred to as a regularization parameter. We have the following result. Theorem 3.1 : The learning operator defined by the minimization of the the performance index (12) has the form L(u(·), x) = Jq∗0 ,T (u(·), x)(κIr + Dq0 ,T (u(·), x))−1 ,
(13)
where Jq∗0 ,T (u(·), x) stands for the adjoint Jacobian (8), and Dq0 ,T (u(·), x) is the dexterity matrix (9). This learning operator coincides with the singularity robust Jacobian inverse (Tcho´ n and Jakubiak 2003, Tcho´ n and Malek 2009), L(u(·), x) = JqSRI (u(·), x). 0 ,T
(14)
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Proof: To begin with, we expand the task space error into the Taylor series, and retain only first order terms, obtaining ek+1 = Kq0 ,T (uk+1 (·), xk+1 ) − yd ∼ = Kq0 ,T (uk (·), xk ) + Jq0 ,T (uk (·), xk )(∆uk (·), ∆xk ) − yd = ek + Jq0 ,T (uk (·), xk )(∆uk (·), ∆xk ), (15) where Jq0 ,T (u(·), x) denotes the Jacobian (5). Now, a suitable substitution from (15) into (12) yields
I(∆uk (·), ∆xk ) = eTk + (Jq0 ,T (uk (·), xk )(∆uk (·), ∆xk ))T (ek + Jq0 ,T (uk (·), xk )(∆uk (·), ∆xk )) + κ||(∆uk (·), ∆xk )||2RW
or, equivalently, I(∆uk (·), ∆xk ) = ||ek ||2 + 2eTk Jq0 ,T (uk (·), xk )(∆uk (·), ∆xk )+ (Jq0 ,T (uk (·), xk )(∆uk (·), ∆xk ))T Jq0 ,T (uk (·), xk )(∆uk (·), ∆xk ) + κ||(∆uk (·), ∆xk )||2RW . Differentiation of I(∆uk (·), ∆xk ) leads to the following expression D I(∆uk (·), ∆xk )(∆vk (·), ∆wk ) =
d |α=0 I(∆uk (·) + α∆vk (·), ∆xk + α∆wk ) = dα
d |α=0 2eTk Jq0 ,T (uk (·), xk )(∆uk (·) + α∆vk (·), ∆xk + α∆wk )+ dα d |α=0 (Jq0 ,T (uk (·), xk )(∆uk (·) + α∆vk (·), ∆xk + α∆wk ))T Jq0 ,T (uk (·), xk )(∆uk (·)+ dα d α∆vk (·), ∆xk + α∆wk ) + |α=0 κ||(∆uk (·) + α∆vk (·), ∆xk ) + α∆wk ||2RW = dα 2eTk Jq0 ,T (uk (·), xk )(∆vk (·), ∆wk )+2(Jq0 ,T (uk (·), xk )(∆uk (·), ∆xk ))T Jq0 ,T (uk (·), xk )(∆vk (·), ∆wk ) + 2κh(∆uk (·), ∆xk )), (∆vk (·), ∆wk )iRW . Now, the condition D I(∆uk (·), ∆xk )(∆vk (·), ∆wk ) = 0 is by (7) equivalent to hJq∗0 ,T (uk (·), xk )ek , (∆vk (·), ∆wk )iRW + hJq∗0 ,T (uk (·), xk )Jq0 ,T (∆uk (·), ∆xk ), (∆vk (·), ∆wk )iRW + κh(∆uk (·), ∆xk ), (∆vk (·), ∆wk )iRW = 0 for every (∆vk (·), ∆wk ), resulting in
−1 ∗ (∆uk (·), ∆xk ) = − κ idX + Jq∗0 ,T (uk (·), xk )Jq0 ,T (uk (·), xk ) Jq0 ,T (uk (·), xk )ek , where idX denotes the identity map on X . But, it is easily seen that (κ idX +Jq∗0 ,T (u(·), x)Jq0 ,T (u(·), x))Jq∗0 ,T (u(·), x)) = Jq∗0 ,T (u(·), x)(κIr +Jq0 ,T (u(·), x)Jq∗0 ,T (u(·), x)),
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therefore, using the definition of the dexterity matrix (9), we conclude that (∆uk (·), ∆xk ) = −Jq∗0 ,T (uk (·), xk ) (κIr + Dq0 ,T (uk (·), xk ))−1 ek = −JqSRI (uk (·), xk )ek , 0 ,T what finishes the proof.
Observe that, somewhat informally, by setting in (13) κ = 0 one obtains the Jacobian pseudo inverse operator for mobile manipulators (Tcho´ n and Jakubiak 2003). The inverse kinematics algorithm based on the iterative learning control scheme with operator (14) will be called the singularity robust Jacobian inverse kinematics algorithm. In continuous setting this algorithm is defined by the dynamic system d (uθ (·), xθ ) = −γJqSRI (uθ (·), xθ )(Kq0 ,T (uθ (·), xθ ) − yd ), 0 ,T dθ
(16)
while the solution of the inverse kinematics problem is computed as the limit of trajectory (uθ (·), xθ ) at θ going to +∞ (Tcho´ n and Jakubiak 2003, Tcho´ n et al. 2010).
4
Convergence assessment
Suppose that the endogenous configurations computed by (16) form a smooth curve cθ = (uθ (·), xθ ) ∈ X , parameterized by θ ∈ R. In practical computations this curve needs to be sampled with a certain rate h, so we let θ = kh. By Theorem 3.1, after introducing the sampling rate, the increments (11) of the endogenous configurations will assume the following form (∆uk,h (·), ∆xk,h ) = −hJq∗0 ,T (uk (·), xk )(κIr + Dq0 ,T (uk (·), xk ))−1 ek = −hJqSRI (uk (·), xk )ek . (17) 0 ,T The following result provides a Kantorovich-type (Deuflhard 2004) estimate for the region of guaranteed convergence of the singularity robust Jacobian inverse kinematics algorithm. Theorem 4.1 : condition
Suppose that the mobile manipulator Jacobian satisfies the following Lipschitz
||(Jq0 ,T (u(·), x) − Jq0 ,T (v(·), w))(u(·) − v(·), x − w)|| ≤ ω||(u(·) − v(·), x − w)||2RW ,
(18)
where || · || denotes the Euclidean norm, ω is a positive constant, and let λ∗ > 0 denote the lower bound of eigenvalues of the dexterity matrix Dq0 ,T (u(·), x). Then, the singularity robust Jacobian inverse kinematics algorithm converges, provided that the initial task space error e0 lies in the open ball B(0, r) ⊂ Rr , of radius 1− r = 8κ
q
1−
ωh2
λ∗ h κ+λ∗
.
(19)
Proof: We begin by computing the error at the step k + 1 ek+1 = Kq0 ,T (uk+1 (·), xk+1 ) − yd = ek + Kq0 ,T (uk+1 (·), xk+1 ) − Kq0 ,T (uk (·), xk ).
(20)
Invoking Proposition 2.4.7 of (Abraham et al. 1988), called the Hadamard’s lemma, we can
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re-write (20) as
ek+1 = ek +
Z1
Jq0 ,T ((uk (·), xk ) + t(∆uk,h (·), ∆xk,h ))(∆uk,h (·), ∆xk,h )dt =
0
ek +
Z1
(Jq0 ,T ((uk (·), xk ) + t(∆uk,h (·), ∆xk,h )) − Jq0 ,T (uk (·), xk )) (∆uk,h (·), ∆xk,h )dt+
0
Jq0 ,T (uk (·), xk )(∆uk,h (·), ∆xk,h ), (21) where Jq0 ,T (u(·), x)(v(·), w) stands for the Jacobian (5). By the Lipschitz condition (18), after a substitution from (17) and exploiting (9), the identity (21) results in
−1
||ek+1 || ≤ ||Ir −hDq0 ,T (uk (·), xk ))(κIr +Dq0 ,T (uk (·), xk ))
||||ek ||+ω
Z1
t||(∆uk,h (·), ∆xk,h )||2RW dt.
0
(22) Now, from the formula (17) we obtain the estimate ||(∆uk,h (·), ∆xk,h )||2RW ≤ h2 ||JqSRI (uk (·), xk )||2 ||ek ||2 . 0 ,T Next, taking into account the assumed bound on eigenvalues of the dexterity matrix, the inequality (22) can be written in the following form ||ek+1 || ≤ α||ek || + β||ek ||2 = (α + β||ek ||)||ek ||,
(23)
where α=
r
1−
λ∗ h , κ + λ∗
(24)
and, since by Lemma 1 of (Tcho´ n and Malek 2009) ||JqSRI (u(·), x)|| ≤ 21 κ−1/2 , we get 0 ,T β=
ωh2 . 8κ
(25)
Finally, using the inequality (23), we deduce that the sequence of error norms ||ek || will converge to zero, if α + β||e0 || < 1, from which, we conclude (19). Furthermore, by the boundedness of JqSRI (u(·), x), the increments of the endogenous configurations (17) will also tend to 0, i.e. 0 ,T lim ||(∆uk,h (·), ∆xk,h )||RW = 0.
k→+∞
(26)
Remark 1 : Essentially, it suffices that the convergence condition proclaimed in Theorem 4.1 holds for all k ≥ N , where N is an integer. This means that for a finite number of steps the algorithm can operate in singular endogenous configurations, in particular, its initial configuration could be singular.
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It turns out that the magnitude of sampling rate h has crucial influence on the size of the region of convergence. Observe that, when the sampling rate decreases to zero, the radius (19) grows up to infinity, and simultaneously, the curve cθ = (uθ (·), xθ ) becomes a trajectory of the dynamical system (16). This conclusion confirms global convergence of the singularity robust Jacobian inverse kinematics algorithm proved in (Tcho´ n and Malek 2009).
5
Computations
In this section we compute the Lipschitz constant (18) and the radius of guaranteed convergence (19) for a stationary manipulator and for a mobile robot. For computational reasons, a finite dimensional representation c = (λ, x) ∈ Rs of the endogenous configuration will be used, where λ ∈ Rs−p denote coefficients of a truncated orthogonal expansion of the control function u(t) (Tcho´ n and Jakubiak 2003). It is easily seen that in finite dimensional case the mobile manipulator Jacobian (5) gets represented by an r × s matrix. The computation of the Lipschitz constant is facilitated by the following lemma. Lemma 5.1: Let Jq0 ,T i (c) denote the ith column of the Jacobian matrix. Suppose that for all i, j = 1, . . . s the norm ||
∂Jq0 ,T i (c) || ≤ δ, ∂cj
(27)
where || · || denotes the Euclidean norm, and δ > 0 is a number. Then, the Lipschitz constant ω = sδ. Proof: For a fixed d ∈ Rs , we denote v = c − d, and define a function F (c) = (Jq0 ,T (c) − Jq0 ,T (d))v. Then F (d) = 0, and by the Hadamard’s lemma we have
F (c) =
Z1
∂F (tc + (1 − t)d) dtv. ∂c
0
Furthermore, ∂F (tc + (1 − t)d) v= ∂c
∂Jq0 ,T (tc + (1 − t)d)v ∂c
v=
s X ∂Jq0 ,T i (tc + (1 − t)d) vi vj . ∂cj
i,j=1
Using the above identities and the estimate (27), we find ||(Jq0 ,T (c) − Jq0 ,T (d))v|| ≤ δ
s X
|vi ||vj | ≤ sδ||v||2 ,
i,j
where the last inequality results from the fact that 2|vi ||vj | ≤ vi2 + vj2 . By comparison with (18) it follows that ω = sδ, and we are done. 5.1
Stationary manipulator
The kinematics of the 3R planar manipulator are defined as k(x) = (l1 c1 + l2 c12 + l3 c123 , l1 s1 + l2 s12 + l3 s123 ),
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where x = (x1 , x2 , x3 ) ∈ R3 is the vector of joint variables, li , i = 1, 2, 3 denote lengths of the links, and ci = cos xi , si = sin xi , etc. The Jacobian is equal to J(x) = [J1 (x), J2 (x), J3 (x)] =
−l1 s1 − l2 s12 − l3 s123 , −l2 s12 − l3 s123 , −l3 s123 . l1 c1 + l2 c12 + l3 c123 , l2 c12 + l3 c123 , l3 c123
are upper bounded by δ = It is easily checked that the norms of all partial derivatives ∂J∂xi (x) j l1 + l2 + l3 , so by Lemma 5.1 the Lipschitz constant can be taken as ω = 3(l1 + l2 + l3 ). Next, assuming that |s3 | ≥ ǫ, what means that the joint positions of the manipulator lie sufficiently far from singular configurations, we compute eigenvalues of the dexterity (in fact, manipulability) matrix D(x) = J(x)J T (x), and discover the following lower bound for the minimal eigenvalue λ∗ =
l22 l32 ǫ2 . (l1 + l2 + l3 )2 + (l2 + l3 )2 + l32
Substitution of these estimates into (19) allows to one compute the radius of convergence. For a manipulator with unit link lengths l1 = l2 = l3 = 1 and at ǫ = 10−3 , figure 1 shows a plot of the radius.
300 250
r1(h,κ)
200 150 100 50 0 0.1 0.2 0.3 h
0.4 0
0.4
0.2
0.6
0.8
1
κ
Figure 1. Plot of r(h, κ) for 3R planar manipulator
5.2
Chained form system
The kinematics of the 3 degree-of-freedom chained form system (Murray et al. 1994) have the representation q˙1 = u1 ,
q˙2 = u2 ,
q˙3 = u1 q2 ,
y = q.
We choose truncated control functions of the form u1 (t) = λ1 + λ2 sin t,
u2 (t) = λ3 + λ4 sin t,
defined over the interval T = 2π. For q0 = 0, the kinematics (2) become K0,T (λ) = 2π(λ1 , λ3 , πλ1 λ3 − λ2 λ3 + λ1 λ4 ),
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whereas the Jacobian
1 0 0 0 0 0 1 0 . J0,T (λ) = [J1 (λ), J2 (λ), J3 (λ), J4 (λ)] = 2π πλ3 + λ4 , −λ3 , πλ1 − λ2 , λ1 Observe that for λ1 = λ3 = 0 the configuration of the chained form system gets singular. Now, i (λ) || ≤ 2π 2 , so the Lipschitz constant can be chosen a computation shows that every norm || ∂J∂λ j as ω = 8π 2 . Furthermore, if λ21 + λ23 ≥ ǫ2 (non-singularity) and λ22 + λ24 ≤ η 2 (boundedness), T (λ) then it turns out that the minimal eigenvalue of the dexterity matrix D0,T (λ) = J0,T (λ)J0,T is lower bounded by λ∗ =
ǫ2 (π 2
ǫ2 . + π + 1) + η 2 (π + 1) + 1
Given κ and h, a substitution of ω and λ∗ into (19) yields the radius of convergence. Under assumption that ǫ = 10−3 and η = 1, figure 2 presents the dependence of the radius from h and κ. In order to illustrate the performance of the singularity robust inverse kinematics algorithm
12 10
r2(h,κ)
8 6 4 2 0 0.1 0.2 0.3 h
0.4 0.5
0
0.2
0.4
0.6
0.8
1
κ
Figure 2. Plot of r(h, κ) for the chained form system
applied to the chained form system, we shall solve 3 example inverse kinematic problem characterized, respectively, by the desirable task space points yd = (1, 1, 1), (−1, 1, −1) and (2, 3, 4). In all these problems the initial control coefficients have been set to λ0 = (0.1, 0.1, 0.1, 0.1), κ = 0.5, h = 0.1. The results are shown in figures 3, 4 and 5 below. The theoretical value of radius r = 5.065060, and the initial task space error ||e0 || = ||K0,T (λ0 ) − yd || whose norms equal to 0.959413, 2.055071, 4.686812, lie within the region of guaranteed convergence. Since convergence is also achieved when the initial error norm exceeds the value of r, our estimate is rather conservative.
6
Conclusion
Within the framework of endogenous configuration space approach we have demonstrated that the singularity robust Jacobian inverse for the mobile manipulators can be defined as an optimal learning operator in the classical iterative learning scheme. This has been done by minimizing a weighted sum of the squared task space error norm and the endogenous configuration norm
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1 0.8 0.6 0.4 0.2 0
||e|| ||e||
0
0.6 0.4 0.2 0 -0.2
20
40
60
80
100
120
140
160
180 200 iterations
u1,u2 u1 u2
0
1
2
3
4
5
t
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Figure 3. Solution to problem 1: task space error and controls
3
||e|| ||e||
2 1 0 0
0.4 0.2 0 -0.2 -0.4 -0.6
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u1,u2 u1 u2
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Figure 4. Solution to problem 2: task space error and controls
6
||e|| ||e||
4 2 0 0
0.8 0.6 0.4 0.2 0
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u1,u2 u1 u2
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Figure 5. Solution to problem 3: task space error and controls
under assumption that the error increment is completely characterized by the mobile manipulator Jacobian. The Jacobian inverse obtained in this way has been employed in a corresponding Jacobian inverse kinematics algorithm. After imposing a Lipschitz condition on the growth of the Jacobian, we have obtained an estimate for the region of guaranteed convergence of the algorithm. Computation of this estimate has been illustrated with the kinematics of a stationary and a mobile robot. Solutions of example inverse kinematic problems for the chained form system confirm validity of the estimate, and suggest that in order to increase the region of convergence
August 10, 2010
15:55
International Journal of Control
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tcon-2010-0201˙rev REFERENCES
one may increase the regularization parameter (at the price of slowing down the speed of convergence) or decrease the sampling rate. Since the estimate appears to be conservative, further research is needed to make it tighter. Another open problem is systematic computation or estimation of the Lipschitz constant in Theorem 4.1. The most challenging research perspective would lead to exploiting learning operators more advanced than the linear operator (13). 7
Acknowledgments
This research has been supported by the Wroclaw University of Technology under a statutory grant. The author is indebted to Dr. Janusz Jakubiak who has kindly accomplished computations presented in section 5. Comments provided by the reviewers are also thankfully acknowledged. References
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