The trajectory planning and control was studied of a robot manipulator that has 6 degrees of freedom and was designed based on the mechanism of the Stewart Platform. First the main components of the manipulator is described along with its operation. The solutions are briefly prescribed for the forward and inverse kinematics of the manipulator. After that, two trajectory planning schemes are developed using the manipulator inverse kinematics to track straight lines and circular paths. Finally experiments conducted to study the performance of the developed planning schemes in tracking a straight line and a circle are presented and discussed

Antrazi, Sami

Nguyen, Charles C.

English

NASA Technical Reports Server

THE CATHOLIC UNIVERSITY OF AMERICA
DEPARTMENT OF ELECTRICAL ENGINEERING
TRAJECTORY PLANNING AND CONTROL
OF A 6 DOF MANIPULATOR WITH
STEWART PLATFORM-BASED MECHANISM
Charles C. Nguyen
Principal Investigator and Associate Professor
and
Sami Antrazi
Graduate Research Assistant
submitted to
Dr. Charles E. Campbell, Jr.
Code 714.1
Goddard Space Flight Center (NASA)
Greenbelt, Maryland
July 1990
https://ntrs.nasa.gov/search.jsp?R=19900016294 2020-03-19T22:36:30+00:00Z
SUMMARY OF THE REPORT
This report presents the research results obtained from the research grant entitled
"Control of Robot Manipulator Compliance, "funded by the Goddard Space Flight Center
(NASA) under a research grant with Grant Number NAG 5-780, for the period between
February 1st, 1990 and July 31, 1990.
This report deals with the trajectory planning and control of a robot manipulator
that has 6 degrees of freedom and was designed based on the mechanism of the Stew-
art Platform. First the main components of the manipulator will be described and its
operation will be explained. We then briefly present the solutions for the forward and
inverse kinematics of the manipulator. After that, two trajectory planning schemes will
be developed using the manipulator inverse kinematics to track straight lines and cir-
cular paths. Finally experiments conducted to study the performance of the developed
planning schemes in tracking a straight line and a circle will be presented and discussed.
1 Introduction
Successful robotic assembly of parts requires that the robot end-effector be able to
perform very precise motion. As a result, research effort has been enormously spent
[1,12] to study the application of closed-kinematic chain (CKC) mechanism in the
design of robot manipulators performing high precision motion because CKC mecha-
nism generally provides better positioning capability than open-kinematic chain (OKC)
mechanism. CKC mechanism was first implemented in the design of the Stewart Plat-
form [1] originally intended for simulating aircraft motion. Later, the Stewart platform
attracted considerable interest of robotic researchers and was proposed in the design
of several robot manipulators and end-effectors [2,8]. Hunt [2,3] developed numerous
structural designs for robot manipulators using the mechanism of the Stewart Platform
which initiated intensive research in [4,5]. Sugimoto and Duffy [6] employed the theory
of linear algebra and screw systems to obtain a general method which describes the
instantaneous link motion of a single closed-loop mechanism. Yang and Lee [7] studied
the feasibility of manipulators designed based on the Stewart Platform whose inverse
dynamics and kinematics were later investigated by Do and Yang [8] and Sugimoto [9].
Nguyen and Pooran applied the Lagrangian formulation to derive dynamical equations
for a six-degree-of-freedom (DOF) CKC manipulator [10] whose kinematics was also
developed in [11]. Later, they derived a learning-based control scheme for the above
manipulator to perform repetitive tasks [12].
A Stewart platform-based manipulator was designed and built at the Goddard Space
Flight Center (GSFC) [13] to serve as a testbed for evaluating the feasibility of au-
tonomous assembly of parts in space. In this report, we first describe the main compo-
nents of the manipulator and explain its operation. We then briefly review the forward
and inverse kinematics of the manipulator. After that, a Cartesian trajectory planning
scheme is developed and employed with a trajectory control scheme using the manip-
ulator inverse kinematics. Finally experiments conducted to study the performance of
the developed trajectory planning and control scheme are presented and discussed.
2 The 6 DOF Robot Manipulator
This section is devoted to briefly describe the main components of the manipulator and
present the solutions for its inverse and forward kinematics.
2.1 Description of Hardware
Figures 1 and 2 show the robot manipulator which was designed based on the Stewart
Platform mechanism and mainly consists of a lower base platform, an upper payload
platform, a compliant platform, a gripper and six linear actuators. The movable payload
platform is supported above the stationary base platform by six axially extensible rods
with ballnuts and ballscrews providing the extensibility. Stepper motors were selected
to drive the ballscrewsto extend or shorten the actuator lengths whosevariations will
in turn produce the motion of the payload platform and consequentlythe motion of a
gripper attached to the compliant platform. Each end of the actuator links is mounted
to the platforms by 2 rotary joints whoseaxesintersect and are perpendicular to each
other. Passivecompliance is provided through the compliant platform, which is sus-
pendedfrom the payloadplatform by six spring-loadedpistons arranged in the Stewart
Platform mechanism.The complianceis passivelyprovided by permitting strain on two
opposingsprings acting in the pistons. Thus the pistons are compressedand extended
when resistive and gravitational forcesare applied on the gripper. The rotation of each
stepper motor is controlled by sendingout proper commandsto an indexer which then
transmits proper pulse sequencesto the stepper motor drive. Therefore, precisegripper
motion can be produced by properly controlling the motions of six manipulator legs.
Figure 2 presents the planning and control schemeemployed to control the motion of
the manipulator gripper. A Cartesian path specifiedwith desired starting and ending
velocities and accelerations is converted into 6 Cartesian trajectories using a Carte-
sian trajectory planning scheme. Then based upon the desired Cartesian trajectories,
joint-space trajectories will be determined by a planner which sends proper commands
through the RS232 port of a personal computer to the indexers. The indexer will then
transmit pulses to the stepper motor drives where microstepping permits each revolution
(360 °) of the stepper motor to be equivalent to 25,000 steps. Therefore the drive rotates
36°° each time it receives onethe stepper motor one angular increment of 25,_ = 0"0144°,
step pulse. Furthermore, through the linear motion converter system consisting of the
ballnut and the ballscrew, each angular increment (=lstep) is converted into 8 #-inches
of linear translation of the manipulator leg.
2.2 Solution of the Inverse Kinematics
The lengths of the manipulator legs are selected as joint variables. To define the Carte-
sian variables we proceed to assign coordinate frame {A} to the movable payload plat-
form and {B} to the base platform as shown in Figure 2. The position and orientation
of Frame {A} with respect to Frame {B} are selected as the Cartesian variables in
the sense that the position of Frame {A} is the position of its origin with respect to
Frame {B}. Now denoting the angle between AAi and xA by )_i, and the angle between
BBi and xB by Ai for i=1,2,... ,6, we have obtain Ai = 60(i - 1)°; hi = 60(i - 1) °
for i=1,3,5 and Ai = Ai-1 + 0B; ,ki = Ai-1 + OA for i =2,4,6. Furthermore, if Vector
Aai = (aix air aiz) T describes the position of the attachment point Ai with respect to
Frame {A}, and Vector Bb i = (bi, biu biz) T the position of the attachment point Bi with
respect to Frame {B}, then they can be written as Aai = [ rAcos(,ki) rasin()_i) 0 ]T
and Bbi = [ rBcos(Ai) rBsin(Ai) 0 ]T for i=1,2,... ,6 where rA represents the radius
of the payload platform and rB that of the base platform.
We proceed to consider the vector diagram for an ith actuator given in Figure 4.
The length vector Bqi = (qi, qiu qiz) T, expressed with respect to Frame {B} can be
computed by
Bqi = Bai - Bbi (1)
where Vector Ba/and Vector Bbi describe the positions of Ai and Bi, respectively both
in terms of Frame {B}. However, Ba i can be computed by
Ba,. = _R aal + Bd (2)
where _R is the matrix representing the orientation of Frame {A} with respect to Frame
{B} and Vector Bd contains the Cartesian coordinates x, y, z of the origin, A of Frame
{A} with respect to Frame {B} such that Bd= [ x y z]T.
Now substituting (2)into (1) and using li = x/q_, + q_ + q_z, we obtain after intensive
simplifications:
li 2 = x 2 + y2 + z 2 + r_ + r_ + 2(r,,ai, + r,2aiy)(x - b,,)
+2(r21aiz + r22ai_)(y -- biy) + 2(r31aiz + r32ai_)z -- 2(xblx), (3)
for i=1,2,... ,6 where rij for i_j=1,2,3 are the elements of the matrix ABR.
Equation (3) represents the solution to the inverse kinematic problem in the sense
that for a given Cartesian configuration, composed of the position and orientation speci-
fied by Bbi and ABR, respectively, the actuator lengths l_ for i=1,2,... ,6, can be computed
using (3).
2.3 Solution of the Forward Kinematics
The forward kinematics concerns with the determination of the Cartesian position and
orientation of the payload platform when the actuator lengths l; for i=1,2,... ,6, are
given. Consequently, the forward kinematic problem can be formulated as to find x, y, z
(position) and r 0 (orientation) for i=j=1,2,3 to satisfy Equation (3) for a given set of li
for i=1,2,... ,6. Thus the problem is reduced to solving 6 highly nonlinear simultaneous
equations with 6 unknowns, 3 of which represent the orientation. A closed-form solution
does not generally exist, which leads us to seek an iterative numerical method to solve
the above set of nonlinear equations. One widely used technique for solving nonlinear
equations is the Newton-Raphson method, which is employed in this report to solve the
forward kinematic problem.
3 The Trajectory Planning Schemes
Two trajectory planning schemes developed to control the motion of the robot manip-
ulator gripper are presented in this section. The first scheme is designed for tracking
straight lines while the second for arbitrary paths.
3
3.1 The Straight-Line Trajectory Planning Scheme
The indexer has two main modes of operation: the normal mode and the continuous
mode. In the normal mode, based on the information about the velocity vl, acceleration
a, and the distance to be traveled At, the indexer will determine the appropriate leg
velocity profiles which are either a trapezoid or a triangle depending on the relationship
between the given information. For simplicity, the straight-line planning scheme will
only utilize the trapezoidal profile which is illustrated in Figure 5 where t_, t_, and
ta denote the acceleration time, the constant velocity time, and the deceleration time,
respectively. In addition the indexer requires that t_ = td. By inspection, we found that
At = v1(tc + td) and v I = ata. To track a path in a 3-dimensional space, the positions
of x- y- and z-coordinates must always be linearly related to each other anytime during
the tracking. Intuitively, if the leg displacements are planned such that their velocities
are linearly related to each other, then the resulting Cartesian motion should be a
linear path. Computer simulation which utilized the manipulator forward kinematics
and was performed to verify the above fact has agreed with our intuition. The following
algorithm facilitates the trajectory planning for straight lines.
Algorithm 1: Straight Line Trajectory Planning
. Use the manipulator inverse kinematics to compute the leg lengths corresponding
to the starting point P8 and the final point P! of the straight line, namely I;8 and
l_f for i=1,2,...,6.
2. Compute Au = lif -- lis for i=1,2,...,6 and find Ark whose absolute value is the
biggest.
. Select ak and vfk for the k-th leg such that ak <_ amax; and v]k <_ vmax; vlk <
ak _-_ to ensure trapezoidal profile where amax and vmax denote the maximum
acceleration and velocity of the stepper motor, and then compute t_ = vlk = ta
ak
andre= v_I - ta.
z_, and vli=Qai.4. For i#k; i=1,2,..,6 compute ai = t_(Q+L)
3.2 The Trajectory Planning Scheme For Arbitrary Curves
In the continuous mode of operation, the indexer requires the acceleration a, final ve-
locity vf and the direction of the rotation (direction of the linear displacement) of the
stepper motor. The stepper motor will accelerate to the velocity v I and continue to run
at this velocity until new velocity and acceleration are given in the same direction of
rotation. The current planning mainly consists of dividing a space curve into n segments
and then planning the velocity profiles of the manipulator legs in the continuous mode
4
so that each segment will be reached within a specified time. The following algorithm
is proposed to plan for arbitrary space curves.
Algorithm 2: Arbitrary Curve Trajectory Planning
1. Divide the space curve into n segments.
, Use the manipulator inverse kinematics to compute the leg lengths corresponding
to each segment point on the curve, namely lij for i=1,2,...,6 (leg number) and
j=1,2,...,n+1 (segment point number)
3. Compute Aij = li,j + 1 - li,j for i=1,2,...,6 and j=l,2,...n.
4. For each segment, select an appropriate travel time tj for j=l,2,...,n, and compute
the corresponding acceleration and final velocity at the end of each segment.
In general, the travel times for the segments are constant and equal to each other
during the tracking of curves which do not require the change of leg direction. However
when direction of any leg has to change, the travel time can be selected efficiently using
the look ahead method. Using this method, the algorithm looks at the next segment
point and determine if any change in leg direction is necessary. For example if the
direction of a leg requires direction change, then its travel time will be recomputed to
ensure that the velocity at the end of the segment will be zero to allow direction change.
After that, the recomputed travel time will be set for the remaining legs for the next two
segments. Finally the travel time of all legs will be set back to the old value before the
leg direction change occurs. The above process can be repeated any time a leg direction
change is necessary.
4 Experimental Study
This section presents the results of experiments conducted to evaluate the performance
of the developed trajectory planning schemes. In particular, Algorithm 1 was used
to track a straight line in the x-y plane of the base frame {B}, which is described
by y(t) = x(t). The required profiles of the manipulator legs required to track the
straight line were determined before the tracking using Algorithm 1 and are illustrated
in Figure 6. The path that the manipulator gripper actually tracked, is presented in
Figure 7 together with the desired path. Figure 7 shows excellent tracking capability
with insignificant errors. Algorithm 2 was applied to compute the leg velocity profiles
shown in Figure 8 to track a circle in the x-y plane of the {B} with a radius of 0.8
inches. Figure 9 shows the desired path plotted together with the path the manipulator
actually tracked. The average errors for x- y- and z-coordinates were computed to be
1.096x10 -4 inches, 2.326x10 -4 inches, and 2.974x10 -5 inches, respectively.
5
5 Conclusion
The problem of trajectory planning for a Stewart Platform-based manipulator was ad-
dressed in this report. We first described the main components of the manipulator and
briefly presented the development of its inverse and forward kinematics. We then em-
ployed the normal and continuous modes of operation of the stepper motor indexer to
derive two trajectory planning schemes for tracking straight paths and circular paths,
respectively. Experiments performed to evaluate the developed algorithms showed ex-
cellent tracking results. Future research will focus on investigating the application of
the algorithms to carry out assembly tasks such as mating space connectors with or
without force and position feedback. In addition, development of a 6 DOF force sensor
system based on the mechanism of the Stewart Platform is under way.
References
[I]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
Stewart, D., "A Platform with Six Degrees of Freedom," Proc. Institute of Mechanical
Engineering, vol. 180, part 1, No. 5, pp. 371-386, 1965-1966.
Hunt, K. H., Kinematic Geometry o.f Mechanisms, Oxford University, London 1978.
Hunt, K. H., "Structural Kinematics of in-parallel-actuated Robot Arms," Trans. ASME,
J. Mech., Transmis., Automa. in Des., Vol. 105, pp. 705-712, 1983.
Fichter, E.F. and MacDowell, E.D., "A Novel Design for a Robot Arm," ASME Int. Com-
puter Technology Conference, San Francisco, pp. 250-256, 1980
Fichter, E.F., "A Stewart Platform-Based Manipulator: General Theory and Practical
Construction," Int. Journal of Robotics Research, pp. 157-182, Summer 1986
Sugimoto, K. and Duffy, J., "Application of Linear Algebra to Screw Systems," Mech.
Mach. Theory, Vol. 17, No. 1, pp. 73-83, 1982.
Yang, D. C. and Lee, T. W., "Feasibility Study of a Platform Type of Robotic Manipulators
from a Kinematic Viewpoint," Trans. ASME Journal of Mechanisms, Transmissions, and
Automation in Design, Vol. 106. pp. 191-198, June 1984.
Do, W.Q.D. and Yang, D.C.tt., "Inverse Dynamics of a Platform Type of Manipulating
Structure," ASME Design Engineering Technical Conference, Columbus, Ohio, pp. 1-9,
1986.
[9]
[10]
Sugimoto, K., "Kinematic and Dynamic Analysis of Parallel Manipulators by Means of Mo-
tor Algebra," ASME Journal of Mechanisms, Transmissions, and Automation in Design,
pp. 1-5, Dec. 1986.
Nguyen, C.C., Pooran, F.J., "Dynamic Analysis of a 6 DOF CKCM Robot End-Effector
for Dual-Arm Telerobot Systems," Journal of Robotics and Autonomous Systems, Vol. 5,
pp. 377-394, 1989.
6
[11] Nguyen, C.C., Pooran, F.J., "Kinematic Analysis and Workspace Determination of a 6
DOF CKCM Robot End-Effector" Journal of Mechanical Technology, Vol. 20, pp. 283-
294, 1989.
[12] Nguyen, C. C., Pooran, F.J., "Learning-Based Control of a Closed-Kinematic Chain Robot
End-Effector Performing Repetitive Tasks", to appear in Journal of Microcomputer Appli-
cations, Vol. 8, No. 3, 1989.
[13] Premack, Timothy et al, "Design and Implementation of a Compliant Robot with Force
Feedback and Strategy Planning Software," NASA Technical Memorandum 86111, 1984.
7
Figure 1: The robot manipulator.
]32 ]33
Figure 2: Frame assignment for platforms.
Space Space Manipu-
Trajectory Trajectory Computer Motor
Planning Planning Indexers Lator .
Desired Desired Joint-Space Commands Pulses Motion
Cartesian Cartesian Trajectories
Path Trajectories
Figure 3: Trajectory planning and control of the robot maafipulator.
ZA
Payload
Platform
IA} A YA
Ai
:1i
B i
V
/ I Slope a Slope
Figure 4: Vector diagram for the ith leg. Figure 5: Trapezoidal velocity profile.
2%
J i 1 i
v3
,, vl ",_
_-;,, ..............._, 4............."?%-,__
, , -'_'--..'_.'
1 2 3 4
Time (see)
2
00
i i i i
Dotted: Actual
I I I t
0.5 1 1.5 2
X (in.)
Figure 6: Leg velocity profiles (Alg. 1).
vertical azis=velocit!t (rev/sec)
Figure 7: Tracking a straight line.
vertical azis=y-azis (inch}
2 v3 !
|, J .;- ........ ..._-_._ .- ...... ,,.
0 -:_
I I I
0 5 10
Time (sec)
2
0
Time (see)
Figure 8: Velocity profiles for 6 legs (Algorithm 2).
vertical azis=velocit!l (revolution/sec)
0.5
0
-0.5
r Dotted: Actual
o: Desired __
_ °o a_,'x9_-, ,2,,e_, ,z,_ ooO °v" _ I
-0.5 0 0.5
X-axis (in.)
Figure 9: Tracking a circular path.


Trajectory planning and control of a 6 DOF manipulator with Stewart platform-based mechanism

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