Robotics and Autonomous Systems - EPFLlasa.epfl.ch/publications/uploadedFiles/monot_optimal_pp3.pdf · tion with more complex geometry. Attempts have also been made to construct a

Robotics and Autonomous Systems 61 (2013) 209–220

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Robotics and Autonomous Systems

journal homepage: www.elsevier.com/locate/robot

A direct variational method for planning monotonically optimal paths forredundant manipulators in constrained workspacesAshwini Shukla, Ekta Singla, Pankaj Wahi, Bhaskar Dasgupta ⇤Department of Mechanical Engineering, Indian Institute of Technology Kanpur, Kanpur-208 016, India

a r t i c l e i n f o

Article history:Received 12 October 2011Received in revised form2 August 2012Accepted 17 August 2012Available online 28 September 2012

Keywords:Redundant manipulatorMonotonic optimalityGlobal path planningObstacle avoidance

a b s t r a c t

This paper proposes a path planner for serial manipulators with a large number of degrees of freedom,working in cluttered workspaces. Based on the variational principles, this approach involves formulatingthe path planning problem as constrained minimization of a functional representing the total jointmovement over the complete path. We use modified boundary conditions at both ends of the trajectoryto find more suitable start and end configurations. The concept of monotonic optimality is introduced inorder to optimize themanipulator paths between the resulting end configurations. For obstacle avoidance,volume and proximity based penalizing schemes are developed and used. The presented planner usesa global approach to search for feasible paths and at the same time involves no pre-processing task. Avariety of test cases have been presented to establish the efficacy of the presented scheme in providinggood quality paths. The extent of advantage accruing out of the measures of free end-configurations andmonotonic optimality are also analyzed quantitatively.

© 2012 Elsevier B.V. All rights reserved.

1. Introduction

Service manipulators in most commercial applications areexpected to perform complicated tasks in cluttered environments.This necessitates the development of techniques for autonomouspath planning in arbitrary 3-dimensional workspaces. In orderto increase the manipulability and dexterity of a manipulatorworking in such environments, it has been widely accepted toincrease the number of links of the manipulator and utilize theextra degrees of freedom (refer to [1–3]). However, it complicatesthe motion planning problem due to increasing complexity of theinverse kinematics of redundant manipulators with each addeddegree of freedom (DOF). In this paper, we address this problemusing a direct variational approach.

The problemof generating collision-free paths formanipulatorswith increasingly large numbers of links has attracted considerableattention in the last two decades [4–8]. Broadly, the strategiesproposed and pursued in this period can be classified as localand global (refer to [9,10]). Local methods start from a giveninitial configuration and step towards the goal configuration usinglocalized information of the workspace. On the other hand, theglobal path planningmethods typically apply search algorithms onthe precomputed connectivity graphs of the free regions of C-space(refer to [11,12]). Some recent works [13,14] present improvised

⇤ Corresponding author. Tel.: +91 512 259 7995; fax: +91 512 259 7408.E-mail address: [email protected] (B. Dasgupta).

methodologies which have been proven better than completelyrandomized techniques.

The local approaches towards path planning involve searchinga grid placed across the robot’s configuration space [15] aroundthe current configuration. Artificial collision measures computedfrompartial information of the geometry of the configuration spaceare used to guide the search, which in most cases turn out to be‘‘greedy’’ in nature. Most proposed heuristics are in the form ofarticulated potential-field functions, guiding the search along theflow of its negative gradient vector field [16–19]. These are knownas the potential field methods. Although these methods are foundto be effective for local obstacle avoidance, in highly constrainedworkspaces, they suffer from the drawback of eventually leadingthe search to dead ends, i.e. local minima of the proposed potentialfield. Moreover, with each added DOF to the manipulator, thenon-linearity of the problem increases progressively, making thetechnique more prone to such dead ends and, in turn, impracticalfor redundant manipulators in highly constrained workspaces. Afew researchers [20–22] have presented neural network basedtechniques for trajectory control of redundant manipulators. Theyemphasize the importance of motion planning for manipulatorswith large numbers of degrees of freedom working in constrainedenvironments.

The global techniques have made this highly complex prob-lem tractable by employing complete1 methods, such as thoseusing Roadmap [23,24] and Cell-Decomposition [25,26], through a

1 An algorithm is called complete if it either finds a solution or reports the non-existence of the same.

0921-8890/$ – see front matter© 2012 Elsevier B.V. All rights reserved.doi:10.1016/j.robot.2012.08.012

210 A. Shukla et al. / Robotics and Autonomous Systems 61 (2013) 209–220

Nomenclature

i: Configuration numbern: Number of configurations (control points)j: Joint numberN: Number of degrees of freedomk: B-spline orderqj(t): j-th joint variableq(t): General configurationpi: i-th control configurationx: Combined vector of control configurations: variable

vector to be optimizedl: Inequality constraint numberm: Equality constraint number.

probabilistic framework. In general, roadmap based methods con-sist of generating a network (called roadmap) of one dimensionalpaths in the configuration space [23,24,27]. Such a roadmap con-sists of free configurations connected to one another. Path plan-ning is thus reduced to searching for a path from the start nodeto the end node. On the other hand, the cell-decomposition meth-ods aim at representing the set of collision-free configurations as acollection of cells and search the graph representing the adjacencyrelation among these cells [25,26]. One basic disadvantage of theC-space based global techniques is that they require an exhaus-tive pre-computation exercise in order to develop the connectivitygraphs. Since the computation time required by this constructionis typically exponential in the dimension ‘N ’ of the configurationspace (refer [28]), the approach is impractical even for reasonablylarge values of ‘N ’.

Recently, some researchers have used the calculus of variationsto develop another class of methods in the global approachto path planning. Cecil and Marthaler [29,30] used level setmethods for minimizing energy-based functionals to get locallyoptimal paths. However, their approach is shown to be applicablefor only 2- and 3-dimensional C-spaces. Dasgupta et al. [31]used action minimization by reducing the functional to Euler’sequations and solving them using the relaxation method. Theyachieved relatively complicated manipulation tasks in higherdimensional C-spaces, but only for planarmanipulators.Moreover,their algorithm requires a feasible initial path which they obtainfrom the probabilistic roadmap (PRM) technique and subsequentlyoptimize over the same. In contrast, our approach in this paperworks for spatial manipulators as effectively as that for planarmanipulators and does not require any feasible initial path.

Trade-off between speed and completeness of local and globaltechniques respectively, led to the development of hybrid meth-ods, utilizing the desirable features of both global and localmethods. Krogh and Thorpe [32] and Tournassoud [33] used ge-ometrical solutions for global planning and potential fields for lo-cal planning in the same planner. However, their applications arelimited tomobile robots in low dimensional C-spaces.Warren [34]considered operating upon the entire path at each iteration, whileusing a repulsive potential on obstacles. The results shown arelimited to planar manipulators and to mobile robots operating in2-D. Barraquand and Latombe [19] presented a potential based ap-proach used in conjunction with Brownian movement (randomwalk) to escape from local minima. While this approach works forsimple geometries in 2-D, it suffers from the need of application-specific potential functions and exhibits performance degrada-tion with more complex geometry. Attempts have also been madeto construct a potential field with only one minimum (refer to[35–38]), but the analytical definition of such a potential turnsout to be mathematically involved and computationally expensive

[36,38], similar in drawback to the construction of a connectivitygraph in global methods. Furthermore, even if a definition is avail-able, the applicability of such methods has only been shown forplanarmanipulators and low-dimensional C-spaces. A recentwork(refer to [39]) presents a stochastic technique for planning pathswith the objectives of maximum smoothness, but ignores the issueof obstacle avoidance. The strategy is expected towork successfullyeven for higher degrees of freedom.

All the aforesaid techniques use prescribed initial and final con-figurations generally computed by a separate inverse kinematicsroutine. However, it is to be noted that when the start and endconfigurations are deemed fixed, a degradation from optimality isinherited by the problem definition itself.

1. The chosen start and goal configurations may not be the bestchoice in terms of the path length in joint space, increasing thetotal joint movement associated with the trajectory.

2. The chosen start and goal configurationsmaynot be at all reach-able from each other.

These issues have been addressed partially by Bertram et al. [40].They have incorporated the calculation of the goal configurationwithin the planner, keeping only the start configuration fixed.It is also shown that a randomly chosen pair of configurationsmay lie in disconnected regions of free C-space, rendering thechosen configurations unreachable from each other. However,their technique remains a local one, building a rapidly-exploringrandom tree (RRT) starting from the provided initial configuration,and hence inherits the ever-present drawbacks of a local searchtechnique discussed earlier.

In this paper, we present a path planner for spatial redundantmanipulators working in arbitrary 3-D environments. The pre-sented method uses a global approach without resorting to theneed for developing the complete C-space connectivity graph. Itinvolves defining an energy-based functional in such a way that itsminimization over the function space of all the possible paths re-sults in the definition of a feasible path, together with the initialand final configurations. A partially random initial guess, whichmay be infeasible, is generated using a proximity based heuris-tic, ensuring that the initial path sweeps over roughly the sameCartesian region as the given start and end points. The problemis formulated in a non-linear optimization scheme and the aug-mented Lagrangian optimization technique of constrained opti-mization (refer to [41,42]) is used to numerically minimize thefunctional. We introduce the concept of monotonically optimalpaths and incorporate a mild penalty term in the original objectivefunction, to improve the quality of the solution obtained. The effi-ciency of the approach in handling spatial manipulators in higherdimensional C-spaces is demonstrated through various case stud-ies in cluttered workspaces.

In the next section, we propose a B-spline formulation forthe manipulator path. We recognize a set of variables, whichcollectively define the complete path in joint space. Section 3elaborates the collision detection and quantification scheme. InSection 4, we formulate the energy based functional and use it as apart of the objective function in the subsequently formulated non-linear programming (NLP) problem. In Section 5, we formulate theoptimization problem and discuss the solution strategy using theaugmented Lagrangian technique. Results have been shown in Sec-tion 6 and a comparison has beenmadewith themethod presentedin an earlierwork. Finally, some conclusions are drawn in Section 7.

2. Path modeling

For amanipulatorwith ‘N ’ joints, a path expressed in joint spaceis a collection of ‘N ’ scalar functions of time (qi(t)), with eachfunction qi representing themotion of the i-th joint in time. Hence,

A. Shukla et al. / Robotics and Autonomous Systems 61 (2013) 209–220 211

the path can be expressed as a vector function of a scalar variable,viz. q(t) as,

q(t) = [q1(t) q2(t) · · · qN(t)]T . (1)

In order to numerically minimize the energy functional (refer toSection 4) over the function space of all paths, it is first requiredto choose an appropriate function model for approximating theabove vector function and identify a set of variables governingthe shape of each of its component function curves. Developingsuch a model requires a set of basis functions, the weighted sumof which would give the function value at any point. Candidatebasis functions which can be used are {1, x, x2, . . .}, Chebychevpolynomials, Bessel functions, B-spline basis functions, etc. (referto [41]).Weuse theB-spline functionmodel in thiswork. Followingconsiderations arising from the inherent properties of B-splines(refer to [43]) make them particularly useful in the current contextand hence justify the choice.

1. For a k-th order B-spline curve, each point on the curve isaffected only by the nearest k control points. This provides localcontrol over the path to bemodeled and, in the current context,leads to effective collision avoidance while maintaining smallerpath lengths.

2. The function is defined piecewise. Therefore, it is possible totrack complicated paths even with low degree B-splines.

3. Start and end points can be interpolated by using open-uniform2 knot vectors. This ensures effective handling of thekinematic constraints (refer to Section 5).

4. B-splines lie within the convex hull created by the correspond-ing control polygons3. This simplifies the evaluation of jointlimit constraints of the manipulator (detailed in Section 5).

We use the de Boor recursive formula [43] to calculate thebasis functions for a B-spline of k-th order. If pi 2 <N , forn + 1 control points (p0, p1, . . . , pn) and n + 1 basis functions(B0,k, B1,k, . . . , Bn,k), the B-spline curve is given by

C(t) =nX

i=0

piBi,k(t). (2)

It is to be noted that each of the control points pi 2 <N repre-sents one control configuration of the manipulator with its com-ponents giving the corresponding joint variable values. We choose‘n+1’ such configurations defining the complete path. Continuousrepresentation of the path for all joints is obtained by using Eq. (2)with these ‘N(n + 1)’ values as control points.

However, we recognize the fact that time is a scalable quantity.The same path when traversed in a different span of time resultsin a different set of functions qi(t). Hence, an infinite numberof function definitions exist for the same path. To avoid suchredundancy, we choose a constant range parameter t 2 [0, 1].It is to be noted here that one may choose different functions⌧ (t) mapping the parameterization from the chosen parameter totime, so as to keep the joint velocities and accelerations withinpermissible limits. The representation of the manipulator path injoint space is now complete and can be expressed as

x ⌘ [pT0 pT

1 pT2 · · · pT

n]T where pi 2 <N , x 2 <N(n+1). (3)

2 Open-uniform knot vectors are the ones with uniform knot spans and k equalknot values at each end. This ensures that the resulting B-spline passes through thefirst and the last control points.3 More specifically, if t 2 [ti, ti+1), then the point corresponding to t on the curve

lies inside the convex hull of control points (pi�k+1, pi�k+2, . . . , pi).

3. Collision detection and quantification

The problem of collision detection in path planning for physicalobjects is central to any model-based manipulation system. In thissection, we present a scheme to calculate a measure of infringe-ment/proximity between the manipulator and the workspace,which is used to generate a penalty function in the variational for-mulation (Section 4) for collision avoidance.

At a given configuration, the manipulator is tested against theworkspace obstacles and among its own links for collisions. Ifa collision is reported, the algorithm computes the approximatevolume of infringement and derives from it a normalized measureof infringement, which is reported as a negative value. On theother hand, if no collision is reported, a normalized shortestdistance between the manipulator and obstacles is reported as apositive value. The detailed discussion is divided into three steps,viz. modeling of the workspace and the manipulator, collisiondetection and penetration volume calculation.

3.1. Object modeling

For proximity analysis of the manipulator with the givenworkspace obstacles, we use their triangularized surface meshes.Any solid modeling software can be used to generate the surfacemesh in the form of a stereolithographic file (.STL). To conformwith the data structures used in the application, each of the objectspresent in the workspace must be introduced in the form ofseparate components placed appropriately. A convenient way todo this is bymaking an assembly of all the obstacles, placed at theirintended positions. We use the Solid-Works Assembly (.SLDASM)format for this purpose. Zhang et al. [44] have presented anddiscussed theirwork on penetration depth computation for convexand non-convex objects. However, no comparison is presented asa part of this paper.

The manipulator links are assumed to be rectangular blocks.No external information is needed for their modeling exceptthe D–H parameters and width of the links as input to theapplication. Links are meshed as collections of bounding cuboidsand nodal coordinates, whenever required, are transformed usingmanipulator kinematics.

3.2. Collision detection

To detect if there is a collision between the manipulator andthe obstacles or within the manipulator links, we use a librarycalled the Proximity Query Package (PQP). This library performsfast proximity queries on a pair of surface triangularized solidmodels using OBB-Trees and swept sphere bounding volumes (BV)as introduced by Gottschalk et al. [45] and Larsen et al. [46]. Theavailable queries in this package are collision detection, separationdistance computation and tolerance limit verification. We first usethe collision detection query which reports whether there is acollision between the input models. If no collision is reported,we invoke the separation distance query which gives the shortestdistance ds between the input solid models. This process isrepeated for all links and obstacle pairs and the reportedminimumdistances for all links are added after normalizationwith respect tothe link lengths. IfN is the number of links and Lj denotes the lengthof link j, proximity measure d is obtained as

d =NX

j=1

ds,jLj

.

Normalization with respect to link length avoids scaling variationsdue to a change in the units.


3.3. Penetration volume calculation

If collision is reported between the manipulator and theobstacles orwithin the links,we calculate the infringement volumeto quantify the amount of collision. We use the 3-dimensionalversion of the point-in-polygon technique given by Nordbeck andRystedt [47] to classify a given point in space as inside or outsidea general polyhedron. This technique, also known as the sum ofangles method, involves computing the signed sum of solid anglessubtended by the faces of a polyhedron at the point of interest. Ifthe sum equals 4⇡ or �4⇡ , the point is reported to be lying insidethe polyhedron, if it equals 0, the point is reported outside. If anyvalue other than these is obtained, the polyhedron is not closedand hence the problem is ill-posed. A robust and generalized 3-D implementation is given by Carvalho and Cavalcanti [48]. Usingthis method, the approximate penetration volume is computed asfollows:

1. Discretize each link into a number of cells in transverse andaxial directions.

2. For each cell, find out the number of its vertices lying inside theobstacle polyhedron.

3. If n vertices are found inside, report n/8 of the cell volume asthe collision volume.

4. Repeat Steps 1–3 for all cells and keep adding the collisionvolumes.

5. Normalize the total infringement volume obtainedwith respectto the total link volume and sum for all links as

d = �NX

j=1

3

vuutV collisionj

V totalj

.

Anerror analysis is presented in theAppendix, to examine the errorin the penetration quantification, done using the methodologyused above. The variations in the error with respect to the changein the penetration represents the conservative nature of theapproach.

4. Variational formulation

The variational approach involves formulating the path plan-ning problem as minimization of a functional. We define thefunctional using different terms representing path length andpenalty due to the manipulator’s proximity or infringement withworkspace obstacles. The final path is obtained as a localminimizerof this functional. The functional is based on the Lagrangian (L) andis defined as explained below.

At a particular value of the parameter s = s⇤ 2 [0, 1], we haveq(s⇤) 2 <N and the Lagrangian

L = 12qTWq (4)

tends to keep the path short. A symmetric positive definite weightmatrix W can be used to counter the scaling variations amongdifferent components of the vector q arising from differences inunits or types of joints. Moreover, W can also be used to assign ahigher weightage to some joints in the process of joint movementminimization to account for different inertia experienced bydifferent joints. In this implementation,we have usedW = I, i.e. alljoint values are kept at the same scale and priority.

The issue of obstacle avoidance is addressed in the form ofa penalty function P(q) introduced as a constraint in the opti-mization formulation (Section 5). It is derived from the infringe-ment/proximitymeasure described in Section 3which ensures thatthe path remains within the free regions of the C-space. To define

P(q(t)), we first calculate the infringementmeasure d correspond-ing to configuration q(t) as explained in Section 3. Now, for somet 2 [0, 1], P(q(t)) is defined as

P(q(t)) = �c1(1 � e�d/c2) (5)

) @P(q(t))@d

= � c1c2

e�d/c2 . (6)

Here, c1 and c2 are respectively associated with the overall scaleof the problem and steep rise of the potential curve with respect tothe infringementmeasure d. Although the gradient of this functionis governed by c1 to some extent, in order to exercise independentcontrol over both properties of the curve, a separate variable c2 isintroduced. Variations of the potential function with c1 and c2 areshown in Fig. 1. In this work we have used c1 = 100 and c2 = 5.

Complete definition of the functional is now obtained by inte-grating the above formulated Lagrangian function over the com-plete path, thereby obtaining a scalar quantity A, which is ameasure of the total path length in joint space:

A =Z 1

0Ldt

=Z 1

0

⇢12qTWq

�dt. (7)

The above definition is closely related to Hamilton’s principle ofleast action with a Lagrangian as defined in Eq. (4). According tothis principle, a dynamic system of one or more bodies evolvesfrom one time instant to another in such a way that the actionremains stationary. Hence, for such systems, invoking first orderoptimality conditions on the action A provides a convenient wayto formulate the differential equations of the path function q(t) inthe form of Euler’s equations [31]. However, the solution of sucha system of differential equations requires the knowledge of theboundary conditions, i.e. in this case, the initial and final configu-rations. These end configurations fixed arbitrarily by the user implycertain drawbacks as explained earlier. In thiswork, instead of gen-erating the Euler equations, we minimize the action numerically,not insisting on fixed end configurations, by directly formulatingan optimization problem as explained in the next section.

5. Optimization scheme

As explained in the previous section, the path planning problemis reduced to the minimization of action A, which is formulated asa function of the manipulator path. In Section 2, it is shown thata manipulator path can be represented completely by ‘N(n + 1)’variables. Hence, the problem can now be treated as a multivariateoptimization problem with basic elements of the objective func-tion and the corresponding solution space completely defined. Thefollowing discussion describes the complete optimization formu-lation and the solution procedure.

We define the problem in a constrained optimization frame-work asminimize f (x)subject to gl(x) 0 for l = 1, 2, 3, . . . , L;and hm(x) = 0 form = 1, 2, 3, . . . ,M.

Here, f (x) is the objective function to be minimized, subject tothe inequality constraints gl(x) and equality constraints hm(x). Thissection define the formulation in detail.

5.1. Objective function

The objective function f (x) to be minimized is the action(Eq. (7)) combined with a mild penalty term. To justify the need


Fig. 1. Potential curves for different parameter values (a) c1 = 1, c2 = 5, (b) c1 = 100, c2 = 5, (c) c1 = 1, c2 = 1, (d) c1 = 1, c2 = 20.

Fig. 2. Monotonic vs non-monotonic paths, insufficient deviation penalty.

of this term, it is first required to introduce the concept andsignificance of monotonically optimal manipulator paths. Of all thepossible paths between the given start and end configurations,a path with monotonic motion on all the joints correspondsto a minimum amount of joint movement. We call such pathsmonotonically optimal. A schematic representation in Fig. 2(a)explains how a monotonic path, represented by the dotted line,incurs a net joint movement which is 2�q less compared to itsnon-monotonic counterpart. These paths are of high significance asthey correspond to the least amount of energy expenditure amongall possible paths between the two fixed configurations. Basedon this heuristic, any deviation from non-monotonicity should bepenalized proportionally by the amount of deviation.

If qj(t) represents the motion of the j-th joint with respect tothe path parameter t 2 [0, 1], then the amount of deviation frommonotonicity is defined as

� =NX

j=1

Z 1

0�j(t)dt,

where

�j(t) =(|qj(t + tstep) � qj(t)|,

if SGN[{qj(t + tstep) � qj(t)}{qj(1) � qj(0)}] < 00 otherwise.

However, this scheme fails to penalize cases where qj(t) exhibitsnear-horizontal slopes, i.e., qj(t + tstep) � qj(t) ⇡ 0. An illustrationof such a case is provided symbolically in Fig. 2(b), where thedeviation from monotonicity is extremely small for the part BCof the curve. Moreover, any small change in the joint motionrepresented by the dotted curve, intended to proceed towardsa monotonic path, will not be allowed since both path lengthand the deviation � increase locally. Therefore, we propose a


much stronger requirement on the joint movement, i.e. linearitywith a specific slope. We penalize any deviations from a desiredslope, which is equal to the slope of the line joining (0, qj(0))and (1, qj(1)). This scheme imposes considerable penalty on allregions of the curve, including regions of zero slope.Moreover, thisrenders the jointmovement close to linearwith respect to the pathparameter and hence ensures smaller joint accelerations and lessoverall torque requirement. The amount of penalty associatedwithslope deviation can be defined as

�s =NX

j=1

Z 1

0�sj (t)dt, (8)

where

�sj (t) =��qj(t + tstep) � qj(t)

tstep� qj(1) � qj(0)

1

�� .

After inclusion of the above penalty term to improve the smooth-ness of the path, we define the objective function as

f (x) =Z 1

0

✓12qTWq

◆dt + �s, (9)

where �s is given by Eq. (8) and is a user-defined scaling pa-rameter ensuring a mild slope deviation penalty. It is to be notedthat the requirement of linearity inherently imposes the require-ment of monotonicity, which ensures smaller path lengths. Hence,incorporation of only�s in the objective function is sufficient.

5.2. Equality constraints

In path planning, equality constraints arise due to the require-ments to be fulfilled by the start and end configurations. In mostof the previous work in this field, including all classes of roadmapand potential methods, fixed configurations are assigned at bothextremes of the trajectory. However, this turns out to be a muchstronger constraint than what is required. The real physical con-straint in most applications is for the end-effector to reach a par-ticular Cartesian location, which is possible with a large numberof configurations, in fact, infinitely many for redundant manipula-tors. Hence, we formulate the equality constraint functions in sucha manner, that instead of penalizing a change in the configuration,they penalize the movement of the end-effector away from theprescribed start and end locations. Since our solution space is de-rived from the joint space of the manipulator (refer to Section 2),we must define a transformation, mapping the joint space to theCartesian space. The transformation between the coordinate sys-tems attached to links j and j + 1 in terms of the D–H parame-ters [49] is given by

T j+1j =

2

664

cos(✓j) � sin(✓j) cos(↵j) sin(✓j) sin(↵j) aj cos(✓j)sin(✓j) cos(✓j) cos(↵j) � cos(✓j) sin(↵j) aj sin(✓j)

0 sin(↵j) cos(↵j) dj0 0 0 1

3

775 .

Assuming frame 0 to be attached with the base of themanipulator,coincident with the world coordinate system and N as the num-ber of links, the transformation from the base to the end-effectorcoordinate frame is given byTN0 = T 1

0 T21 T

32 · · · TN

N�1.

Hence, at a configuration q4 2 <N , the generalized coordinates ofthe end-effector in a Cartesian system are given by�Xtip, Ytip, Ztip, 1

T = TN0 (q) {Xlocal, Ylocal, Zlocal, 1}T ,

4 Where qj represents ✓j or dj corresponding to the type of the j-th joint asrevolute or prismatic, respectively.

where (Xlocal, Ylocal, Zlocal) gives the position of the hand in the end-effector link. Based on this relation, for the start and end configu-rations qstart and qend, we formulate the six equality constraints.

5.3. Inequality constraints

Inequality constraints arise due to two reasons: bounds overthe joint variables and the need of collision avoidance. Higher andlower limits for each joint are prescribed and obtained as inputto the optimization routine. For the j-th joint, these limits can bewritten as qlj qj quj where qlj and quj represent the lower andupper bounds respectively, giving rise to the inequality constraintsas

qj(t) � quj 0 8t 2 [0, 1]and qlj � qj(t) 0 8t 2 [0, 1],for j = 1, 2, 3, . . . ,N where ‘N ’ is the number of joints.

However, this scheme requiresmonitoring of the joint variablescontinuously over the complete path, which is impractical andhighly inefficient. Instead, we use the convex hull property ofB-splines and monitor only the control values corresponding toeach joint. Since a B-spline curve lies completely inside the convexhull of its control points, forcing the control values of a particularjoint to be inside the bounds is sufficient to render the completepath feasible with respect to these inequality constraints. It is tobe noted that the control values for each joint can be directlyextracted from the solution vector as explained in Section 2. If[pT

0 pT1 · · · pT

n]T represents the solution vector at the currentiteration and ‘n + 1’ is the number of control points, then theinequality constraints due to joint limits are given as

(pi)j � quj 0 i = 0, 1, . . . , n

qlj � (pi)j 0 i = 0, 1, . . . , n,(10)

for j = 1, 2, 3, . . . ,N where ‘N ’ is the number of joints.The other inequality constraint arises due to the requirement

of non-colliding paths. It is observed that a point in the solutionspace which is a local minimum with respect to the action A, maynot correspond to a non-colliding path. To enforce such a condition,we introduce another inequality constraint as

gobs = maxt

{P(q(t))} 0, (11)

i.e., the maximum potential energy associated with any configura-tion over the complete path should be less than the colliding limit.

5.4. Augmented Lagrangian method

We now present the solution scheme using the augmentedLagrangian method [50,51]. In a highly constrained problem suchas the one presented in this work, it is difficult for any algorithmto maneuver through the feasible regions of the solution space.However, the augmented Lagrangian method allows non-feasibleiterates in the intermediate iterations, and slowly converges overa feasible and optimal solution. Moreover, there is no need to finda feasible initial guess and hence the choice of this method isjustified.

In this method, the constrained problem is reduced to uncon-strained minimization of the augmented Lagrangian function de-fined as

F(x) = f (x) +LX

l=1

µlgl(x) +MX

m=1

�mhm(x)

+ 12D

LX

l=1

[max(0, gl(x))]2 + 12D

MX

m=1

h2m(x), (12)


Fig. 3. 2-D workspace with 3 obstacles. (a) 2-D path. (b) Variation of minimum distance of approach betweenmanipulator and workspace obstacles along the path. (c) Jointcurvature comparison. (d) Joint motion using Method I. (e) Joint motion using Method II.

where µl and �m are the Lagrange multipliers corresponding tothe l-th inequality and m-th equality constraints respectively. Dis the penalty parameter specified by the user, large enough torender the augmented Lagrangian function convex through theiterations. At each step, the augmented Lagrangian function F(x)is minimized with respect to the primal variables x, at the end ofwhich themultipliersµl and �m (dual variables) are updated. Uponconvergence of the iterations with respect to the primal as wellas dual variables, a feasible and optimal solution is obtained. Fordetails of the method, refer to [41,42].

6. Results and comparisons

The path planning scheme presented in this paper is imple-mented in C++ and tested with various workspaces, so that effec-tive comparisons and conclusions can be made with respect to thedesired properties of the path. The comparisons suggest a two-waydecrease in the path length mainly due to

1. the algorithm choosing better initial and final configurations asagainst the imposed user defined configurations, and

2. optimizing the path completely between the chosen configura-tions, i.e. achieving monotonic optimality.

6.1. 2-D workspace

We present the first case study with a planar 8-DOF manipula-tor working in a 2-D workspace consisting of 3 obstacles as shownin Fig. 3. The link lengths of the manipulator are [8, 8, 7, 7, 6, 6,6, 6] units with base point at (0, 0, 0). A path is planned betweenthe Cartesian points (40, 30, 0) and (30, 10, 0) using the presentedplanner (Method I). This is one of the tasks planned in Dasguptaet al. [31] (Method II).

In order to compare the quality of the paths obtained from thetwomethods, path length and smoothness comparisons have beenmade. In the following discussion, a decrease in path length fromMethod II to Method I due to the two aforementioned factors hasbeen identified separately. Here, the optimal path length refers tothe path length of a monotonically optimal path having the samestart and end configurations as the actual path. It has been used tocalculate the contribution of monotonic optimality in decreasing


Table 1Path length analysis for 2-D workspace.

Path length Optimal path length %DecreaseMethod I (rad) Method II (rad) Method I (rad) Method II (rad) Choice of end configurations (%) Monotonic optimality (%) Total (%)

1.040 4.161 1.040 2.603 37.56 37.99 75.55

Table 2Path length analysis for staggered obstacles.


3.208 8.807 3.208 5.466 25.64 37.93 63.57

Table 3Path length analysis for shelves.


1.849 5.201 1.849 2.085 4.53 59.91 64.45

the path length. To compare the smoothness of the paths wepresent the variation of curvature in joint space, defined as,

(t) =��q00(t) � �

u · q00(t)�u��

||q0(t)||2 , (13)

where u is the unit vector in the direction of q0(t) (refer to [41]). Toestablish the feasibility of the paths obtained in each case, a plot ofthe minimum distance d between the manipulator and workspaceobstacles with respect to path parameter t is shown. A positivevalue of d confirms the non-infringement at all configurationsalong the path. Fig. 3(a) shows the set of configurations obtained bythe presented path planner and the path traced by the end-effectorin Cartesian space. The configurations shown in bold represent thecontrol configurations for the B-spline interpolation.

Fig. 3(b) shows the variation of minimum distance of themanipulator from the obstaclesminj{ds,j}, with the path parametert . Clearly, the value is positive all along the path which confirmsthat the path is devoid of any collision. Fig. 3(c) shows a comparisonof curvature of the path in joint space. The solid line represents thevariation of with path parameter t for the path obtained fromMethod I, whereas the dashed line represents the same forMethodII. It is clear from the plot that the path obtained from the presentedplanner corresponds to a smaller value of curvature in joint spaceand hence lesser joint accelerations and torques. The motion ofeach joint obtained by using Methods I and II is plotted in Fig. 3(d)and (e) respectively.

It is clear from Fig. 3(d) that the path given by the presentedplanner is monotonically optimal. Moreover, for the path obtainedby using fixed end configurations, the optimal path length betweenthe chosen fixed configurations (2.603 rad) is still larger than thepath length obtained from the presented algorithm (1.040 rad).Hence, a part of the total decrease registered (75.55%) in the pathlength is due to the choice of end configurations (37.56%) and therest is due to monotonic optimality of the path between thoseconfigurations (37.99%), as depicted in Table 1.

6.2. 3-D workspaces

6.2.1. Staggered obstaclesWe present another case study with an 8-DOF spatial manip-

ulator working in a 3-D workspace composed of 10 cubical boxesplaced randomly in space. The path is planned between the Carte-sian points (4, �4, 3) and (5, 0, �5). A similar analysis of the path

obtained is given in Fig. 4, as earlier. Fig. 4(a) shows the workspaceand thepath obtained. The variation ofminimumdistance betweenthe manipulator and obstacles with the path parameter t is pre-sented in Fig. 4(b), which clearly shows that there is no collisionbetween the manipulator and the workspace obstacles anywherealong the path. It is evident from Fig. 4(d) that the path given by thepresented planner is monotonically optimal. A comparative anal-ysis with respect to path length is given in Table 2, according towhich the total jointmovement incurred in the path obtained fromMethod I is 63.57% less than that obtained from Method II.

6.2.2. ShelvesThe next testworkspace consists of an array of shelves as shown

in Fig. 5(a). The start and end points are chosen in such away that atypical pick and place operation between two shelves is simulated.Fig. 5(b) shows theminimum distance plot along the path. Fig. 5(c)clearly shows that the curvature of the joint paths obtained fromMethod I are considerably lower than those obtained usingMethodII. The joint paths shown in Fig. 5(d)–(e) show that the pathobtained from the presented method is monotonically optimal asagainst the path obtained fromMethod II, which is highly deviatedfrom the same. Detailed path length analysis is given in Table 3which shows an overall decrease of 64.45% in the path length.

6.2.3. BarsThisworkspace is specially designed to test the algorithm’s per-

formance inmaneuvering through highly constrainedworkspaces.It is a network of crossing bars making a hair-comb like struc-ture. The 6-DOF manipulator is required to start from the point(1.2, �0.5, 2.5) on one side of the structure and pass to theother side to reach the goal point (6, �5, 2.5). Fig. 6(a) showsthe workspace and the final path that was obtained. Clearly, themanipulator navigates through the network of bars without col-liding as depicted by the plot in Fig. 6(b). Fig. 6(c) shows the curva-ture plot and Fig. 6(d) shows the joint motion corresponding to alljoints.

7. Conclusion

A variational calculus based global path planner for spatiallyredundant manipulators is presented in this paper. The approachinvolves defining an energy based functional in the form of a pathintegral, representing the total joint movement. The augmentedLagrangian technique has been used for constrained minimization


Fig. 4. Staggered obstacles with an 8-link manipulator. (a) 3-D path. (b) Variation of minimum distance of approach between manipulator and workspace obstacles alongthe path. (c) Joint curvature comparison. (d) Joint motion using Method I. (e) Joint motion using Method II.

of this functional, resulting in good quality manipulator paths. AB-spline function model has been used for the representation ofpaths. This provided local control over the trajectory, proved tobe efficient in constraint handling and resulted in smoother jointmovement. Use of semi-fixed boundary conditions and insisting onmonotonically optimal paths provided better quality paths withrespect to path length. An improved collision detection schemeusing tools of proximity and infringement volume calculationhas been introduced in order to avoid using numerically largepenalty values. Results for several 2-D and 3-D workspaces havebeen presented which demonstrate the efficiency of the approach.Comparisons with an earlier variational approach have been madewhich suggest a two-waydecrease in path length due to semi-fixedboundary conditions and monotonic optimality of paths.

Appendix. Error analysis

We use a simple illustrative example to calculate the errorbetween the actual and estimated penetration distance betweentwo objects. We choose the two intersecting bodies as a cuboid,

representing a link of the robot, and a sphere, representing anobstacle. Such a case makes it possible to calculate the penetrationvolume in closed form without making it trivial. To recall, thepenetration distance is positivewhen there is no collision, it is zerowhen the surfaces are in contact and negative in collision.

We use the actual penetration distance dact as the independentvariable and calculate its estimate dest using the infringementvolume. The setup is shown in Fig. 7(a) where a cuboid of sizew⇥w⇥h penetrates into a sphere of radius R (with 2R > h > w inthis case) along the Z axis. The infringement volume for differentpenetration values can be calculated as the piecewise continuousfunction

Vinfringe =

8>>>>>>>><

>>>>>>>>:

V1 if |dact| < R �pR2 � w2/4,

V2 if R �pR2 � w2/4 |dact| R �

pR2 � w2/2,

V3 if R �pR2 � w2/2 < |dact| < h,

V4 if h |dact| h + R �pR2 � w2/4,

V5 if h + R �pR2 � w2/4

< |dact| < h + R �pR2 � w2/2,

V6 otherwise;


Fig. 5. Shelves with a 9-link manipulator. (a) 3-D path. (b) Variation of minimum distance of approach between manipulator and workspace obstacles along the path. (c)Joint curvature comparison. (d) Joint motion using Method I. (e) Joint motion using Method II.

where

V1 =Z p

k

�pk

Z pk�x2

�p

k�x2

⇣pR2 � x2 � y2 � R + |dact|

⌘dydx,

V2 =Z p

k�(w/2)2

�p

k�(w/2)2

Z w/2

�w/2

⇣pR2 � x2 � y2 � R + |dact|

⌘dydx

+ 2Z w/2

pk�(w/2)2

Z pk�x2

�p

k�x2

⇣pR2 � x2 � y2 � R + |dact|

⌘dydx,

V3 =Z w/2

�w/2

Z w/2

�w/2

⇣pR2 � x2 � y2 � R + |dact|

⌘dydx,

V4 =Z w/2

�w/2

Z w/2

�w/2

⇣pR2 � x2 � y2 � R + |dact|

⌘dydx

�Z p

k0

�pk0

Z pk0�x2

�p

k0�x2

⇣pR2 � x2 � y2 � R + |dact| � h

⌘dydx,

V5 =Z w/2

�w/2

Z w/2

�w/2

⇣pR2 � x2 � y2 � R + |dact|

⌘dydx

�Z p

k0�(w/2)2

�p

k0�(w/2)2

Z w/2

�w/2

⇣pR2 � x2 � y2 � R + |dact| � h

⌘dydx,

� 2Z w/2

pk0�(w/2)2

Z pk0�x2

�p

k0�x2

⇣pR2 � x2 � y2

� R + |dact| � h⌘dy dx

V6 = w2h;and k = R2�(R� |dact|)2, k0 = R2�(R� |dact|+h)2. The estimatedpenetration distance is then calculated as dest = � 3

pVinfringe.

Fig. 7(b) shows the variation of dest with dact. At the onsetof collision, the above estimation scheme gives a value that islarger in magnitude than the actual penetration. Note that suchan over-estimate is crucial to our methodology in that it createsa steeper penalty function which in turn inhibits the augmented


Fig. 6. Bars with a 6-link manipulator. (a) 3-D path. (b) Variation of minimum distance of approach between manipulator and workspace obstacles along the path. (c) Jointmotion using Method I. (d) Joint curvature plot.

Fig. 7. (a) 3-D illustrative example for error analysis of the penetration distance estimation. (b) Relationship between estimated and actual penetration distances, dest anddact.

Lagrangian iterations from going into the obstacles. However, thecomputationally cheap estimate that is being used here does notremain conservative at higher depths of penetration as can benoticed from the segment of the plot above the diagonal. Sincethe current paper is concerned with lower extents of penetrationand the main algorithm itself is designed to avoid penetration, thisestimate is sufficient for the purposes of this paper.

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Ashwini Shukla received his bachelor’s and master’sdegrees in Mech. Engg. from IIT Kanpur, India in 2009.He is currently pursuing his Ph.D. at EPFL, Lausanne,Switzerland. His research interests include robotics,dynamical systems, etc.

Ekta Singla is currently working as Assistant Professorat the Indian Institute of Technology Ropar, India. Shereceived her Masters of Engineering degree in January,2002, from Thapar University, Patiala, India and her Ph.D.degree from IIT Kanpur, India in September 2010. Sheworked as a lecturer at Thapar University, Patiala fromJanuary 2002 to July 2004 and as a post-doctoral fellowat Université Pierre et Marie Curie, Paris from July 2009 toJune 2010. Her areas of interest are robotic design, robotpath planning, optimization techniques and computeraided design.

Pankaj Wahi received his bachelor’s degree in Mech.Engg. from Guru Ghasidas University, Bilaspur, India in2001 and his Ph.D. degree from the Indian Instituteof Science, Bangalore, India in 2006. He pursued post-doctoral fellowships at TU Berlin during Feb 2005–May2006 and at Texas A&M University during Nov 2006–June2007. Since Sept 2007 he is working as an assistantprofessor at the Indian Institute of Technology, Kanpur,India. His teaching and research interests include robotics,nonlinear dynamics, control systems, etc.

Bhaskar Dasgupta received his bachelor’s degree inMech. Engg. from Patna University, Patna, India (1991),his master’s degree in Mech. Engg. from the IndianInstitute of Science, Bangalore, India (1993) and his Ph.D.from the Indian Institute of Science, Bangalore, India(1997). Since 1997, he is teaching at the Indian Instituteof Technology, Kanpur, India in the Department ofMechanical Engineering, where currently he is holding theposition of Professor. His teaching and research interestsinclude robotics, CAD, optimization, applied mathematicsand computational biology.

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Robotics and Autonomous Systems - EPFLlasa.epfl.ch/publications/uploadedFiles/monot_optimal_pp3.pdf · tion with more complex geometry. Attempts have also been made to construct a