Redundancy Resolution in Minimum-time Path Tracking of Robotic

Manipulators

Alexander Reiter, Hubert Gattringer and Andreas M¨uller

Institute of Robotics, Johannes Kepler University Linz, Altenberger Straße 69, Linz, Austria

Keywords:

Robotics, Optimal Control, Trajectory Planning, Redundant Robots, Inverse Kinematics.

Abstract:

Minimum-time trajectories for applications where a geometric path is followed by a kinematically redundant

robot’s end-effector may yield economical improvements in many cases compared to conventional manipula-

tors. While for non-redundant robots the problem of ﬁnding such trajectories has been solved, the redundant

case has not been treated exhaustively. In this contribution, the problem is split into two interlaced parts:

inverse kinematics and trajectory optimization. In a direct optimization approach, the inverse kinematics

problem is solved numerically at each time point. Therein, the manupulator’s kinematic redundancy is ex-

ploited by introducing scaled nullspace basis vectors of the Jacobian of differential velocities. The scaling

factors for each time point are decision variables, thus the inverse kinematics is solved optimally w.r.t. the

trajectory optimization goal, i.e. minimizing end time. The effectiveness of the presented method is shown by

means of the example of a planar 4R manipulator with two redundant degrees of freedom.

1 INTRODUCTION

In industrial applications such as painting, welding or

gluing a geometric end-effector path is deﬁned leav-

ing only the problem of ﬁnding a suitable time evolu-

tion of the joints of the executing robot. Introduc-

ing minimum-time trajectories may yield economi-

cal advantages as a shorter trajectory duration results

in a lower task cycle time. Kinematically redun-

dant manipulators provide favorable properties such

as increased workspace dexterity and improved task-

speciﬁc adaptiveness compared to conventional, non-

redundant manipulators. The topic of time optimal

trajectory planning has been studied in a large num-

ber of publications. While for non-redundant ma-

nipulators this problem has been solved, e.g. (Bo-

brow et al., 1985; Shin and McKay, 1985; Pfeif-

fer and Johanni, 1986), for redundant robots no sat-

isfying methods have been proposed yet. Concepts

of minimum-time trajectory planning for kinemati-

cally redundant manipulators can be largely separated

into two method families: joint space and workspace-

based techniques. Members of the former group as-

sume, that the path following problem is solved as an

equality constraint to the trajectory optimization pro-

cess. Alternatively, a joint space parametrization is

assumed to be available (Pham, 2014). In the lat-

ter group, methods incorporate inverse kinematics.

Due to the mathematical representation of redundant

robots’ kinematics, solutions are often obtained nu-

merically (Li´egeois, 1977). Redundancy allows to

augment such solutions by adding objectives such as

maximizing performance measures, e.g. directional

dynamic manipulability in (Chiacchio, 1990). How-

ever, the choice of a performance measure is crucial

as it must act as a local proxy for the superseding time

minimization. Methods relying on joint space decom-

position (Wampler, 1987) offer computation of ana-

lytic inverse kinematics for certain manipulator struc-

tures. There are various joint space decomposition ap-

proaches, some have drawbacks such as the inability

to process certain paths, or boundary conditions, c.f.

(Ma and Watanabe, 2004). Others rely on diffeomor-

phisms that may be difﬁcult to obtain, c.f. (Galicki,

2000). Summarizing, the unsolved problem is often a

kinematic one, particularly the redundancy resolution

in the inverse kinematics problem poses difﬁculties.

In Section 2 of this paper, the problem of com-

puting minimum-time joint trajectories for tracking

a kinematically redundant serial manipulator’s pre-

scribed end-effector path is formulated. Section 3

discusses methods to fulﬁll the path tracking con-

straints. The main contribution of this paper is pre-

sented in Section 4. A method is introduced wherein

the path following requirement is treated using an

inverse kinematics scheme underlying the trajectory

Reiter, A., Gattringer, H. and Müller, A.

Redundancy Resolution in Minimum-time Path Tracking of Robotic Manipulators.

DOI: 10.5220/0005975800610068

In Proceedings of the 13th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2016) - Volume 2, pages 61-68

ISBN: 978-989-758-198-4

optimization problem. The numerical inverse kine-

matics scheme is augmented by optimally scaled

nullspace basis vectors of the instantaneous differen-

tial velocity Jacobian in order to obtain time-optimal

trajectories. This new approach is applied to the ex-

ample of a planar manipulator with two redundant de-

grees of freedom in Section 5. Therein, the trajectory

planning problem is formulated using direct multiple

shooting (Bock and Plitt, 1984) and solved using a

modern interior-point method. Unfavorable proper-

ties of time integration and appropriate countermea-

sures are also discussed. Section 6 concludes this con-

tribution and gives insight in possible enhancements

to the proposed method.

The algorithms are presented using an accelera-

tion level inverse kinematics for the sake of brevity

but can be similarly formulated for higher-order

derivatives.

2 PROBLEM DESCRIPTION

2.1 Kinematically Redundant

Manipulators

The conﬁgurationof a robotic manipulatoris uniquely

deﬁned using coordinates q

,i = 1... n in the conﬁgu-

ration space V

, i.e. q

⊤

= (q

,...,q

) ∈ V

. The end-

effector pose z

can be described with its Cartesian

position r

∈ R

and its orientation, denoted by the

rotation matrix R

, i.e. z

= (R

) ∈ SO(3) × R

instead of SE(3). The forward (or direct) kinematics

mapping f : V

7→ SO(3) × R

maps joint conﬁgura-

tions to end-effector poses.

The robot’s workspace W is given as the im-

age of the direct kinematics mapping f, i.e. W =

{C ∈ imf|h(q) ≤ 0} ⊂ SO(3) × R

wherein only

geometrically admissible conﬁgurations (inequality

constraints h) are considered. The workspace dimen-

sion is given as m = dimW.

A serial manipulator is considered kinematically

redundant if n > m, i.e. the conﬁguration space is

of higher dimension than the workspace. For non-

redundant manipulators (where n = m holds) the in-

verse mapping of the forward kinematics, f

−1

, is

well-deﬁned, and in special cases evengiven in closed

form, e.g. for standard 6R robots with a spherical

wrist. In the case of redundant robots, one has to re-

sort to different, mostly numerical or iterative, meth-

ods.

2.2 Optimal Trajectory Planning for

Prescribed End-effector Paths

The goal of minimum-time trajectory optimization

is to ﬁnd the shortest possible time evolution of the

considered manipulator’s joint positions such that a

given end-effector path is tracked while being re-

stricted to technological limitations. In the follow-

ing it is assumed that the path is given by means of

a series of desired poses, continuously parametrized

with a scalar path parameter s ∈ [0, 1], i.e. z

E,d

(s) =

E,d

(s),r

E,d

(s)) : R 7→ SO(3) × R

. The index d

denotes desired quantities. This yields a non-linear

optimization problem (NLP) of the form

min

1dt (1)

s.t. M(q)

q+ g(q,

q) = Q (2)

min

≤ q ≤ q

max

(3)

min

≤

q ≤

max

(4)

min

≤

q ≤

max

(5)

min

≤ Q ≤ Q

max

(6)

0 ≤ s ≤ 1 (7)

s(0) = 0,s(t

) = 1 (8)

˙s ≥ 0 (9)

E,d

(s) = f (q) (10)

q(0) = q

,q(t

) = q

(11)

q(0) =

q(t

) =

(12)

wherein x represents the vector of optimization vari-

ables describing the time evolution of the joint posi-

tions q. Declarations of dependencies of x will be

omitted below. The optimization problem is subjected

to the manipulator’s (in general non-linear) dynamics

denoted as the equations of motion (2) with the vec-

tor of minimal coordinates q and its time derivatives

q and

q. M is the system’s inertia matrix, g represents

non-linear terms in the equations of motion, consist-

ing of the Coriolis, centrifugal, gravitational and dis-

sipative effects. Q indicates the vector of generalized

forces and torques. Limitations of the manipulator’s

joint positions in (3), joint velocities (4) and possi-

bly higher derivatives such as joint accelerations (5)

may also be incorporated. Further bounds are applied

to the generalized forces Q in (6). The desired end-

effector path is prescribed using a monotonically in-

creasing (9), bounded (7) path parameter. (10) repre-

sents the aforementioned path tracking requirement.

Typically, there are also initial and ﬁnal (11), (12)

constraints of the robot’s joint positions and their time

derivatives. In addition, constraints for cyclic tasks

can be formulated as q

= q

and

ICINCO 2016 - 13th International Conference on Informatics in Control, Automation and Robotics

This optimization problem can be decomposed

into the trajectory optimization problem and an un-

derlying path-tracking subproblem.

3 PATH FOLLOWING

In a minimum-time path tracking optimization prob-

lem, the path following constraint can be imposed as

an equality constraint (10). Other approaches assume

a known joint space parametrization to be known,

e.g. (Pham, 2014). Alternatively, an inverse kine-

matics mapping f

−1

: W 7→ V

can be applied to ob-

tain joint quantities from workspace quantities. As

mentioned in Section 2.1, no closed-form solution

to the inverse kinematics problem exists for kine-

matically redundant manipulators. However, there

are other approaches such as joint space decompo-

sition (Wampler, 1987; Ma and Watanabe, 2004)

or Jacobian-based numeric methods (Whitney, 1969;

Li´egeois, 1977).

3.1 Path Following and Inverse

Kinematics

In order to resolve the path following requirement us-

ing inverse kinematics, divide-and-conquer as well

as unite-and-conquer methods can be applied. Joint

space decomposition can be used as a divide-and-

conquer type approach. Therein, a manipulator’s

structure has to be explicitly separated into two or

more parts. Then the inverse kinematics problem

can be solved based on a loop closure condition.

Joint space decomposition makes direct use of kine-

matic redundancy as operations are performed on

joint level. The choice of decomposition may not be

straight-forward and thus a result of an superseding

integer program. The inverse kinematics solution can

be performed analytically only in cases with suitable

geometry but not in general. Also, the enforcement

of the aforementioned loop closure condition is non-

trivial.

In unite-and-conquer methods such as differen-

tial inverse kinematics, ﬁrstly introduced in (Whit-

ney, 1969), a least-squares solution (w.r.t. an end-

effector error quantity) yields all joint quantities at

once. However, this family of methods needs to

be augmented in order to exploit kinematical redun-

dancy. In this paper, the latter type of inverse kine-

matics methods is used.

The derivations of differential inverse kinematics

schemes below are well-known but reproduced here

as an introduction to and a motivation for the main

contribution of this paper presented in Section 4.

The most simple case is ﬁrst-order differential in-

verse kinematics,

= J(q)

q, (13)

wherein J =

∂

∈ R

m×n

denotes the forward kine-

matics Jacobian, a non-square, wide matrix. Thus it

is not invertible, but an approximate solution for the

joint velocities

q can be computed minimizing the er-

ror in the least-squares sense, i.e.

q = J

E,d

(14)

wherein J

= J

⊤



⊤



−1

denotes the right

Moore-Penrose pseudoinverse. Alternatively,

the dynamically consistent pseudoinverse

= M

−1

⊤



−1

⊤



−1

can be used (Khatib,

1988). Compared to (13), for (14) the index d was

added to the end-effector velocity as it is now a given,

desired quantity. For computing the matrix inverse



⊤



in singular conﬁgurations, a regularization

term can be introduced, i.e. J

= J

⊤



⊤

+ κI



−1

with a small κ > 0. In general, non-singular conﬁg-

urations the nullspace of J has dimension n − m > 0,

i.e. the manipulator is capable of internal motion

that does not affect the end-effector motion. This

property can be exploited using an inverse kinematics

scheme (Li´egeois, 1977) that is augmented to pursue

additional goals. Scalar performance measures w

such as kinematic manipulability (Yoshikawa, 1985b)

or dynamic (Yoshikawa, 1985a) manipulability can

be maximized by adding a velocity term to (14).

This velocity points in the direction of v =

∂w

∂q

and is

projected into the nullspace of the Jacobian, i.e.

q = J

E,d

+ Nv (15)

with the nullspace projector N = (I− JJ

). I denotes

the identity matrix of appropriate size. Substituting

(15) in (13) shows that no end-effector motion re-

sults from the additional term. Pose-dependent per-

formance measures such as kinematic or dynamic ma-

nipulability suffer from the fact that they only rep-

resent a local, instantaneous property. As a result,

they can hardly be exploited in the course of an super-

seding trajectory optimization problem minimizing a

global property such as a trajectory time.

Similar inverse kinematics approaches can be set

up for higher time derivatives simply by deriving (13)

w.r.t. time and isolating the highest time derivative

of the joint positions q, e.g. an acceleration-level ap-

proach yields

q = J



E,d

−



+ Nv (16)

wherein v can again represent a performance measure

gradient projected into the Jacobian nullspace.

Redundancy Resolution in Minimum-time Path Tracking of Robotic Manipulators

3.2 Numerical Inverse Kinematics in

Trajectory Planning

In trajectory planning tasks, constraints regarding

derivatives of the joint positions q may be imposed,

e.g. zero joint velocities at the end-effector ﬁnal po-

sition. By close examination of (16), it can be found

that such a constraint is not necessarily fulﬁlled as a

vector v may not have the appropriate magnitude to

cancel all internal accelerations. Thus, a scaling fac-

tor γ needs to be introduced to fulﬁll such require-

ments, i.e. for the case of an acceleration-level ap-

proach

q = J



E,d

−



+ γNv. (17)

The nullspace scaling factor γ introduced in (17)

needs to vary over time in order to stick to multiple

constraints across the trajectory, i.e. γ = γ(t).

4 NULLSPACE BASIS SCALING

Using the above methods, the joint state is changed

such that a performance measure w is maximized lo-

cally, i.e. following the instantaneous gradient

∂w

∂q

projected into the current Jacobian nullspace. Even

if the step size γ is adjusted properly, this may not

yield an optimal joint state evolution across the path.

If a manipulator provides more than one redundant

degree of freedom, the projection of the gradient will

always lie in a subspace of the nullspace. To make use

of remaining free nullspace directions, task priority-

based methods (Nakamura et al., 1987) can be used to

pursue additional (ideally non-conﬂicting) goals with

lower priorities.

The key idea of the present approach is to com-

pute a basis for the Jacobian nullspace, i.e. kerJ =

span{a

},i = 1,.. .,(n − m). The basis vectors a

are

then scaled by factors γ

,i = 1,..., (n− m) and added

to the inverse kinematics solution (17), i.e.

q = J



E,d

−



n−m

∑

i=1

(t)a

. (18)

For manipulators with a kinematic redundancy of

n − m = 1, there is only one basis vector of the

nullspace. Thus, redundancy is fully exploited by

both approaches, performance measure-based meth-

ods and nullspace basis scaling. However, for higher

degrees of redundancy n − m > 1, exploiting the full

nullspace as in (18) enables an superseding optimiza-

tion process to directly modify the joint trajectories

according to the criteria to be minimized. In contrast

to other approaches, there is no need of a projected

x in m

y in m

0.5

1.5

2.5

−1

−0.5

0.5

Figure 1: Planar manipulator with four revolute joints in

initial conﬁguration.

performance measure gradient acting as a proxy func-

tion to pursue the optimization goals. As for both,

performance measure gradients and the nullspace ba-

sis vectors, symbolic expressions can be obtained be-

forehand by means of computer algebra systems, the

difference in computational cost is negligible.

Section 5 shows that this approach can be easily

applied to problems with readily available kinematic

and dynamical models.

In the method development (13) to (18), only po-

sition coordinates were treated as workspace coordi-

nates for simplicity. Adding a prescribed end-effector

orientation increases the complexity of the problem.

The differential inverse kinematics needs also to be

computed for the end-effector’s angular velocity or its

time derivatives. Care has to be taken in order to es-

tablish consistency in the physical units of products

of the Jacobian.

5 EXAMPLE

5.1 Kinematic and Dynamic Model

The method presented in Section 4 is illustrated us-

ing the simple example of the planar manipulator de-

picted in Figure 1, moving along straight line paths.

The robot consists of four revolute joints (4R), its

links have masses m

L,i

and moments of inertia C

L,i

(about their respectivecenters of mass). The joints are

directly actuated by means of motors without mass

and inertia. Damping coefﬁcients d

account for joint

friction. The system is not inﬂuenced by gravity. Nu-

merical values used for the simulation can be obtained

from Table 1.

A minimal coordinate representation of the ma-

nipulator’s conﬁguration yields q

⊤

= (q

) ∈

ICINCO 2016 - 13th International Conference on Informatics in Control, Automation and Robotics

Table 1: Parameters of planar manipulator.

symbol description value

L,i

link mass 10 kg

link length 1 m

L,i

link moment of inertia m

L,i

/12

damping coefﬁcient 0.1 Nm/rad

i,max/min

joint position limits ±π rad

˙q

i,max/min

joint velocity limits ±2 rad/s

i,max/min

joint torque limits ±10 Nm

V = S

wherein S represents the 0-sphere of each

joint’s admissible range of positions (q

i,min

i,max

The forward kinematics mapping can be easily de-

rived by hand and is given by f. The equations of mo-

tion M(q)

q+g(q,

q) = Q can be derived using well-

known techniques such as the LAGRANGE formalism

or the Projection Equation (Bremer, 1988). The vec-

tor of generalized forces Q

⊤

= (M

) con-

sists of the motor torques.

In this example, the end-effector position, but

not its orientation is considered as workspace coordi-

nates, i.e. r

⊤

= (x, y) ∈ W = {C ∈ imf|q ∈ V} ⊂ R

Thus, the degree of kinematic redundancy is n− m =

5.2 Inverse Kinematics

Applying (18) to the present example yields the in-

verse kinematics law

q = J



E,d

−



∑

i=1

(t)a

. (19)

The inverse kinematics laws (19) and above have only

instantaneous, point-wise characteristics. In order to

obtain joint trajectories for a given end-effector path,

(19) has to be evaluated for all

E,d

along the path

and time integrated using numerical methods to ob-

tain lower time derivatives, i.e.

q →

q =

qdt →

q =

qdt. (20)

Numerical time integration introduces workspace

drift errors e = r

E,d

− f (

q),

e =

E,d

−J(

q. This is-

sue can be avoided by adding stabilizing terms to (19)

such that

e+ K

e+K

e = 0. Re-writing the error dy-

namics in terms of single-order ordinary differential

equations by e

= e and e

e yields







0 I

−K





(21)

whose structure can be exploited for pole-placement.

Incorporating the error dynamics scheme (21) into

(19) yields

q = J



E,d

−

q+ K

e+ K



∑

i=1

(t)a

. (22)

In the pseudoinverse J

= J

⊤



⊤

+ κI



−1

regular-

ization is conducted with κ = 10

−10

. The matrices K

and K

were chosen such that all poles of the error dy-

namics are at −2. In this simple case, the nullspace

basis vectors a

∈ kerJ(q) can be computed analyti-

cally as functions of the system’s parameters and the

current joint conﬁguration.

5.3 Task

For this example, the robot’s task is to move its end-

effector along straight line paths

E,d

= r

+ sL



cosϕ

sinϕ



(23)

of length L = 1 m wherein s ∈ [0,1] denotes the path

parameter. The task is to be performed for slopes

ϕ = 0, π/4, π/2,.. .,7π/4, c.f. Figure 1. At the ini-

tial point r

⊤

= (2,0) m the manipulator’s conﬁgura-

tion is chosen to be

⊤

= (π/3,−2π/3, 0,2π/3) rad.

The task is performed as a minimum-time rest-to-rest

maneuver, i.e.

q(t = 0) =

q(t = t

) = 0.

5.4 Direct Multiple Shooting Trajectory

Optimization

In this section, the optimization problem posed in

Section 2.2 is reformulated incorporatingthe speciﬁcs

of the used manipulator from Section 5.1, its inverse

kinematics scheme from Section 5.2 and the task de-

ﬁned in Section 5.3. Using direct multiple shooting,

originally developed in (Bock and Plitt, 1984), the

time domain is discretized into N uniform intervals

and scaled with the ﬁnal trajectory time t

as a de-

cision variable. In each interval [t

k+1

] the system’s

state is integrated using a fourth-order explicit Runge-

Kutta scheme, denoted as the function f(x

),u

Further declaration of dependencies of t

are omit-

ted for the sake of brevity. Therein the optimization

system’s state consists of the path parameter and its

time derivative as well as the manipulator’s joint po-

sitions and velocities, i.e. x

⊤



, ˙s

⊤



. u

⊤

( ¨s

,γ

k,1

,γ

k,2

) represents the control input of the opti-

mization problem as piecewise constant functions.

The vector of decision variables x consists of con-

catenations of the intermediate states x

,k = 0,.. .,N

and the controls u

,k = 0,...,N − 1, as well as of the

trajectory’s ﬁnal time t

, i.e.

⊤



⊤

,...,x

⊤

N−1

⊤

N−1

⊤



. (24)

Redundancy Resolution in Minimum-time Path Tracking of Robotic Manipulators

The NLP posed in Section 2.2 can be thus reformu-

lated as

min

(25)

s.t. 0 ≤ s

≤ 1 (26)

˙s

≥ 0 (27)

min

≤ q

max

(28)

min

≤

max

(29)

= 0,s

= 1 (30)

˙s

= ˙s

= 0 (31)

(32)

= 0 (33)

= J



E,d

) −

+ K

) +

∑

i=1

i,k

(34)

min

≤ M(q

)

+ g(q

) ≤ Q

max

(35)

k+1

− f(x

) = 0 (36)

wherein kerJ

= span



i,k



. The state is constrained

by (26) to (29). There are also initial and ﬁnal condi-

tions of the state, (30) to (33), but the ﬁnal joint posi-

tions q

are free and obtained as a result of the NLP.

Equation (34) describes the inverse kinematics reso-

lution law used in the state integration and the com-

putation of the inverse dynamics constrained by (35).

As the NLP is implemented as direct multiple shoot-

ing, (36) is required to close the state integration gaps

between adjacent shooting intervals.

For direct multiple shooting, the structure of the

Jacobian matrix of the (in)equality constraints w.r.t.

the decision variables x is blockdiagonal if the order-

ing of x

and u

is as described in (24). Block diago-

nal matrices enable efﬁcient solution algorithms to be

applied. However, in this special case where the ﬁnal

time t

is also a decision variable, an additional dense

column reduces the matrix sparsity as all constraints

depend on this variable. The corresponding sparsity

pattern for N = 5 is depicted in Figure 2.

As this NLP is non-convex,globally optimal solu-

tions cannot be guaranteed. Solutions obtained from

this problem depend on the initial guess. For the ini-

tial guess of the path parameter s

, a linear time evo-

lution was assumed, resulting in a constant velocity

proﬁle for ˙s

. For these s

and ˙s

, an initial guess for

the time evolution of the joint positions q

is obtained

using an numerical inverse kinematics scheme on ve-

locity level. The controls u

are initialized with zeros.

For efﬁcient numerical solution the NLP was im-

plemented using MATLAB interface to the optimiza-

tion framework Casadi 3.0rc3 (Andersson, 2013) and

solved with Ipopt 3.12.3 (HSL MA27 for linear sub-

problems).

Figure 2: Sparsity pattern of the constraint Jacobian for N =

5 uniform time intervals.

The trajectories obtained by the method described

above are continuously differentiable once w.r.t. time,

i.e. q(t) ∈ C

as a second-order inverse kinemat-

ics resolution scheme with piecewise constant inputs

was used. The method can be easily generalized

to higher levels of continuity by simply deriving the

inverse kinematics scheme (22) and adding further

states to the NLP. This is useful for system that require

continuous torques (q(t) ∈ C

), especially differen-

tially ﬂat elastic systems (q(t) ∈ C

), c.f. (Springer

et al., 2013).

5.5 Results

The optimization problem posed in the previous sec-

tion was solved with a time discretization of N = 100

uniform intervals. As an example, the obtained trajec-

tories for ϕ = π/2 are shown in Figure 3 for the path

parameter, and in Figure 4 and Figure 5 for the joint

positions and velocities, respectively. The results sat-

isfy the constraints posed in (26) to (36) and provide

a minimum for t

. However, this minimum is only lo-

cal as the optimization problem is non-convex. The

resulting joint torques are depicted in Figure 6. It can

be seen that at all times, at least two of the constrained

joint velocities or joint torques are saturated, which is

one of the characteristics of a minimum-time property

of the obtained trajectories. Corresponding snapshots

of the manipulator’s optimal motion are depicted in

Figure 7.

Iteration counts, computation times obtained us-

ing an INTEL XEON E3-1246 V3 processor as well

as the resulting trajectory end times for all slopes ϕ

can be obtained from Table 2.

ICINCO 2016 - 13th International Conference on Informatics in Control, Automation and Robotics

˙s

¨s

time in s

path parameter, vel., acc.

0.5

1.5

−1

−0.5

0.5

1.5

Figure 3: Path parameter s, velocity ˙s, acceleration ¨s (ϕ =

π/2).

time in s

joint angles in rad

0.5

1.5

−3

−2

−1

Figure 4: Joint positions q (ϕ = π/2).

˙q

time in s

joint velocities in rad/s

0.5

1.5

−1.5

−1

−0.5

0.5

1.5

Figure 5: Joint velocities

q (ϕ = π/2).

time in s

Motor torques in Nm

0.5

1.5

−10

−5

Figure 6: Motor torques Q (ϕ = π/2).

6 CONCLUSION

This study has presented a contribution to the solution

of the time-optimal path following problem for kine-

matically redundant manipulators. In this approach,

the problem is divided into the trajectory optimiza-

tion and an underlying inverse kinematics problem.

x in m

y in m

0.5

1.5 2

−1

−0.5

0.5

Figure 7: Time evolution along the path with ϕ = π/2 (10

snapshots equally distributed in time).

Table 2: Optimization results for all ϕ.

ϕ t

in s #it. CPU time in s

0 2.1262 206 29

π/4 2.4746 290 41

π/2 2.4073 206 30

3π/4 1.7304 213 30

π 1.7583 157 22

5π/4 2.5625 222 31

3π/2 1.5966 239 34

7π/4 1.1505 134 19

The former is solved using a numerical computation

scheme, augmented to fully exploit redundancy in an

optimal way such that the latter problem yields op-

timal results. It was discussed that this method is

valuable for robots with multiple redundant degrees

of freedom.

The method was successfully applied to a pla-

nar manipulator with two redundant joints moving its

end-effector along prescribed straight line paths.

In future work, the method proposed in this pa-

per will be applied to more complex, spatial examples

also incorporating prescribed orientations. Regard-

ing the implementation, the multiple shooting method

from Section 5.4 can be reﬁned to use non-uniform

shooting intervals to be included as decision variables

instead of varying t

and using uniform time intervals.

This would allow for local adjustment of the time res-

olution and will yield a purely blockdiagonalstructure

in the Jacobian of constraints, c.f. Figure 2.

ACKNOWLEDGEMENTS

This work has been supported by the Austrian

COMET-K2 program of the Linz Center of Mecha-

Redundancy Resolution in Minimum-time Path Tracking of Robotic Manipulators

tronics (LCM), and was funded by the Austrian fed-

eral government and the federal state of Upper Aus-

tria.

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