Acta Astrosautica Vol. 37. pp. 6986. 199s Elsevier Science Ltd. Printed in Qreat Britain
0094~5765(95)000488 PATH
PLANNING
TO A CLASS
WITH
OF SPACE F. Karray*,
OBSTACLE BASED
AVOIDANCE
FLEXIBLE
AS APPLIED
MANIPULATORS
t
V. J. Modi **; and J. K. Ghan***
Department of Mechanical Engineering The University of British Columbia Vancouver, B.C., Canada, V6T 124
.
Abstract
troductiog
A major design concern in the area of space robotics pertains to manipulator systems with a payload r& tio significantly higher than that achieved on ground. This implies structural flexibility of the manipulator links. The design study of such systems presents a formidable problem particularly at the dynamics and the control levels. Besides the stringent performance specifications, the designer is required to account for the environment in which the manipulator operates. Obviously, this would involve obstacle avoidance in the workspace of the system. One distinctive feature of the proposed Space Station, intended to be operational in the late 99’s, will be the Mobile Servicing System (MSS). This multilink flexible manipulator will assist in construction, operation, maintenance and future evolution of the Space Station. It will help in loading and unloading of cargo on board the Space Shuttle, positioning of payloads for specific missions, launching of spacecraft, retrieval of disabled satellites for repair and numerous other tasks. Extending over KKlm, the Space Station is a highly flexible structure with the fundamental frequency in bending of around 0.1 Hz and closely spaced harmonicsl. Due to its size, weight saving requirement and microgravity working environment, the manipulator is also inherently flexible. The interactive dynamics involving the Space Station and the MSS is challenging as it involves relative slewing and translational motion of the flexible manipulator on a highly flexible platform.
The present study is a natural extension to the earlier work carried out by the authors in the area of nonlinear control of flexible structures using the Feedback Linearization Ttique (FLT). This procedure accounts for the complete nonlinear dynamics of the system. It develops a rather innovative approach to path planning, for a alink flexible manipulator, with a specified target and several arbitrarily positioned obstacles. It is important to point out that the preeence of link vibrations makes the problem at least an order of magnitude more complicated rendering the conventional algorithms developed for rigid systems virtually of little use. The problem involves generation of regions (contours) representing spheres of influence around the finite size obstacles, with various degrees of permissible penetrations during path planning to the target. To begin with, equations of motion of an orbiting flexible manipulator, undergoing planar slewing maneuvers, are obtained accounting for a shift in the center of mass during maneuvers. Next, a composite control procedure, involving the FLT together with active vibration suppression using piezoelectric actuators, is developed which promises high degree of tracking accuracy. Depending on the performance requirement of the mission, penalty functions are assigned for a given performance index pertaining to the flexible dynamics of the systems. This permits global improvement of the manipulator operation. Finally, effectiveness of the collision avoidance scheme is illustrated through several examples of increasing complexities.
The present study builds on earlier contributions by the authors2e3 in the area of nonlinear control of orbiting flexible manipulator systems using the Feedback Linearization Technique (FLT). It is a rather versatile and powerful tools being able to account for the complete nonlinear dynamics of the system. It preeents an attractive approach to path planning, for an orbiting 2&k flexible manipulator, with a specified target and several arbitrarily positioned obstaclea The presence of link vibrations makee the problem rather chaBenging as the conventional algorithms developed for rigid systems are generally unsuitable. The method adopted involves generation of regions
t Investigation supported by the Networks of Centers of Excellence Program, Grant No. IRIS/C+ 555380 and Natural Sciences and Engineering Reeearch Council of Canada, Grant No. A2181. * Reeearch Associate.. ** Profmr, Fellow AIAA, Academy Member IAA. *** tiearch Engineer, Ontario Hydro. Copyright @ 1994 by the authors. Published by the AIAA, Inc. with permission. Released to IAF/AIAA to publish in all forms.
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(contours) representing spheres of influence around the finite size obstacles, with various degrees of permissible penetrations during path planning to the target. This makes it possible to adopt modified versions of several well known techniques for path generation with obstacle avoidance4. The equations of motion for an orbiting flexible manipulator, during inplane operation, are obtained accounting for a shift in the center of mass during maneuvers’. Next, smooth trajectories for the manipulator joints, accounting for obstacle avoidance, are derived. Attention is now turned to the FLT3, together with an active vibration suppression control scheme using piezoelectric actuators, which promise a high degree of tracking performance. Depending on the objectives of the mission, penalty functions can be assigned for a desired performance index, which depends on the flexible dynamics of the system. This results in significant performance enhancement of the manipulator operation on the global scale. A set of numerical simulation results are presented to illustrate validity of both the modified obstacle avoidance scheme and the composite control procedure. Integration of a computer vision module to allow for the shape recognition of obstacles is attractive, however, it would demand a significant increase in the computational cost and time. 2.
Dynamical
7: space platform, mobile base, joint 1, link 1, joint 2, link 2, and the payload. Directly attached to the mobile base is joint 1 to which link 1 is connected through an elastic and dissipative transmission unit. Joint 2 and link 2 are similarly connected at the tip of link 1. The payload is considered as an extension of the gripper and both are treated together as one rigid body. The system is free to undergo planar librational or pitch motion about its center of mass, which describes a circular orbit around the Earth. In the case of large maneuvers with a massive payload, the system center of maSs can shift significantly. The formulation accounts for such a shift in the center of mass due to slewing and translational maneuvers. However, the contribution of flexibility on the shift is found to be small’ and hence neglected. All maneuvers and elastic deformations are also confined to the orbital plane. The mobile base, considered rigid, can traverse the Space Station executing translational as well as rotational maneuvers. Joints 1 and 2 can perform slew maneuvers to direct the MSS to any desired orientation thus positioning the payload as required. ‘Danslation &(t) and rotation S*(t) of the mobile base are treated as specified coordinates consistent with the constraints of the physical system. The rotation of link i at joint i denoted by the generalized coordinate r~,, are given by 7~~ and r~~ for joint 1 and joint 2, respectively.
Formulation
A relatively general dynamic model of the MSS accounting for structural flexibility of the links, joints, as well as the supporting platform is considered. The development of the dynamical formulation involves statement of the mathematical model used, derivation of the kinematic relations, evaluation of the kinetic and potential energies, and finally application of the Lagrangian procedure to obtain the governing equations of motion6.
The two flexible manipulator links, treated as EulerBernoulli beams, are free to deform transversely in the orbital plane. The discretization of the flexural deformations is accomplished using assumed modes in conjunction with generalized time dependent coordinates. Thus transverse deflections
2.1
Model
of link
1: CL, (z, t) = g&r,(z)6~,~(t):
Geometry
A relatively simple yet effective dynamical model of the Space Station based flexible MSS system is shown in Figure 1. Despite the Space Station being flexible as stated, it is purposely assumed to be rigid in this study to focus attention on the effect of the MSS’s structural and ,joint Rexibilities during obstacle avoidance maneuvers. The model treats the Space Station as an arbitrarily shaped rigid body in the gravity gradient orientation while the MSS is modelled as a two link robot mounted on a mobile base. The system consists of seven structural members, referred to as bodies 1 to
j=l
deflection
of link 2: [h(z,t)
= ~~~j(~)6L,j(t). j=l
Expressing the 61,~~ and 6~~~ collectively as vectors _ 6~~ and g~s of dimensions nr and np, respectively, the deflections can be written as: . T&r (2, t) = 6L, (t) +1(s); T&.,(I, t) = 6~~0) &(a); where Jr(z) shapes. 2.2
and k(z)
Equations
are vectors
of Motion
containing
the mode
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Using the appropriate kinematics, and proceeding in a manner similar to that outlined in 6, the kinetic (T) and potential (II) energies of the system are readily obtained. Following the Lagrangian approach, the governing equations of motion can be obtained from ar+av=fJj, aqj aqj
j=
l,...,n,
(1) where qj and Qj are the generalized coordinates and the associated generalized forces, respectively. The generalized coordinates are {@,r.rr, 6~,i, TJ,, 6L2i) representing system pitch;
+6L,i
?Jl(t) 6Lzi
7J#
3.2

generalized coordinate i’h mode of link 1;

slew, joint 1;

generalized coordinate associated with the ilh mode of link 2;

slew, joint 2.
The specified coordinates dent maneuver are:
&(t)  translation @b(t)  rotation
representing
time depen
of the mobile base;
of the mobile base.
Mh t)ii + F(q,4, t) = Q, where q and Q are vectors of the generalized coordinates and generalized forces, respectively. The mass matrix M is symmetric and positive definite. It is a function of both q and t due to the nonlinear and time varying nature of the system inertia. The force vector F includes all other forces not associated with the acceleration of q, such as coriolis and centrifugal. Structural damping can be included quite readily as a generalized force. Obstacle
Avoidance
The following highlights obstacle avoidance in the presence of manipulator redundancy. To begin with, manipulator’s redundancy is explained. Next the obstacle avoidance problem is analyzed using a set of mathematical tools based on the notion of redundancy. The concept of obstacle avoidance with a sphere of influence, to account for manipulators structural flexibility, is also discussed. 3.1
Notion of Manipulator’s
Hierarchical
Tasking Approach
associated with the
The resulting nonlinear, nonautonomous and coupled set of equations of motion can be cast in the form
3.
A manipulator is said to have redundancy if it has more degrees of freedom than are necessary to perform a given task. Redundancy allows for higher versatility and broader applicability. Redundant manipulators (RM) have the potential to avoid singularities, as well as obstacles and enable manipulators to reach even behind objects. RM are also reliable in the sense that they can perform tasks even after frailure of some joints. This is an important feature that is quite relevant in space and deep sea applications. RM are nevertheless complex in structure, bulkier and necessitate powerful control alghorithms to deal with positioning and orientation tasks.
Redundancy
The complexity of tasks assigned to a manipulator can be significantly reduced using the task decomposition approach in a hierarchical manner. This approach is based on dividing a given task into several subtasks with a priority order. Each subtask is performed using the degrees of freedom that remain after all the subtasks with higher priority have been imple mented. The task of tracking a trajectory through a workspace with obstacles might be decomposed into controlling the position and orientation of the end efrector and avoiding obstacles, with the former given the first priority. Similarly, control problems of redundant manipulators can be approached through the hierarchical subtasking: the main objective of trajectory tracking may be considered as the subtask with the first priority; and regarding the reason for using the redundancy (obstacle avoidance, singularity avoidance) as the subtask with the second priority. Each subtask may be described in terms of either a desired trajectory, or a certain criterion. For example, Let us consider here the case of a task decomposed into two subtasks, where the first subtask is given in the form of tracking a desired trajectory. 3.3
Mathematical
Formulation
Let the manipulator have n degrees of freedom. The variable associated with the ith joint is q,. The manipulator configuration is denoted by the vector q = (ql, qz, q,)T. Let the first subtask be described by an ml dimensional vector Yr, which is a function of q, y1=
h(9).
(3)
Similarly, the second subtask is described by an mzdimensional vector ys, which is also a function of q, Ya = G(q). (4) The problem now is to find first the joint velocity that achieves the first subtask of realizing yr = Yrd (de sired yr) and then perform as much of the second task
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as possible. The general solution of the joint velocity that performs the first subtask is given by 4= J& + (I  J:Jdh (5) where J1 is the Jacobian with respect to ~1, and kl is an ndimensional arbitrary constant vector. The superscript + denotes the pseudoinverse of the corresponding matrix. The second term on the righthand side of the the previous equation represents the redundancy left after performing the first subtask’. The second subtask is specified by the desired trajectory ysd of the manipulator variable yo. The main idea here is to replace, after some mathematical manipulations, the constant vector kr by the right quantity in order to realize ysd as close as possible. It can be shown’ that the desired joint velocity vector qd when the two subtasks are achieved is given by b = J:hd + (I J:JdHdm  ~a), (6) where Hs is an appropriate diagonal gain matrix suggesting whether the redundancy is used or not. 3.4
Sphere of Influence
To account for the disturbances induced by the structural flexibility of the manipulator links, a zone around the obstacle is introduced. Its avoidance permits determination of the end effector trajectory while accounting for the vibration of the links. Notice that the size of this wne, referred to as the sphere of influence, strongly depends on the structural parmeters of the links as well as on the prescribed trajectory characteristics of the joints. It is of course possible to generate endeffector paths that can completely avoid the interior of these spheres of influence. But this comes at the expense of smaller manipulability of the system. As shown in the simulations results, a combination of the FLT with an active vibration suppression allows for avoidance of the sphere of influence without reducing the capability of the manipulator.
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tant inputs for the FLT, are computed by means of the strategy developed in Section 3.3. When combined with an active vibration scheme, the FLT is expected to provide good joint tracking performance with obstacle avoidance, while keeping a relatively large manipulability region. 4.1
FLT for MIMO Systems
Consider a nonlinear MEMO afBne control systems delined in the local coordinates x E R”, u E R”‘, y E RP for the state space, control space and output space, respectively, with p < m. Using the governing of mo tion and the mapping ic = f(x) + g(x)u ; (7) Y=
h(x) i
where f E Cw(R”; R”) is a smooth vector field with component functions (11, .. . , I,,) and g = (gl, a, g,,,), gi E COD(Rn,Rn). h f C=(R”;RP) is a smooth mapping with component functions h = (hr, . a, II,,). Here R denotes the space of the real numbers and Cm stands for infinitely differentiable mapping. Recall that the Lie Derivative of a function V E P(R”, R) with respect to the vector field f above is defined to be the function LfV : R” + R given by
with LJV denoting the kth iterated Lie derivative. Te Lie bracket of the vector fields X = (Xl,  .  , X,) : R” ) R”, Y = (Yl,. . +, Y,,) : R” + R” h the vector field [X,Y) = (Zr,..., Z,) with
(9) Each multiindex k = (kl, me., kp) E NJ’, N denoting the natural numbers, and each state x E R” define the matrix L,,L>_$(.) . *. L,L:‘_%I(.) E(x; k) =
4.
Nonlinear
Control
For the purpose of dealing with the nonlinear control aspect of the system under study, it would be appropriate to summarize relevant aspects of the mathematical theory of the feedback linearization technique. The approach is based on the recent developments in the area of differential geometry as applied to a class of affine control systems and is outlined in the work of Isidori’. This is an attractive method given that the nonlinear system, as represented by eq. (2), can be readily transformed into an affine form. The notions of aero dynamics and relative degree of a given system are also outlined. Following the mathematical review, the FLT is applied to the model described earlier. The commanded joint trajectories, which are also impor
( L,, LY‘hp(.)
. i.
L*LF‘h,(.)
I (10)
in. Mpxm(R). 4.2
Relative Degres and Normal Form
The system in eq. (10) is said to have a relative degree vector r = (rr, . . . , tp) E NP at a state x0 E R” iE k) = 0 for every multiindex k 4 r (i.e. 1 < ki < ti, 1 5 i 5 p), with x in the neighborhood of x0; and
6) E(x;
(ii) E(x”; r) is of full rank, E(x”; r) = p. n(x”) is the maximal neighborhood of x0 for which E(x; k) = 0 and rankE(x;r) = pforeveryk
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tl(xO).
has nonzero Jacobian on Q(x’).
Now, to obtain the normal form of the nonlinear affine system, the strategy is to apply a change of state and control variables: u ‘Q(X) + P(x)v;
(11)
2 =0(x);
(12)
so that the transformed 5 =Wx)V(x)
Now eq. (7) can be transformed usingeqs. (11),(12), with o, fl and @ defined in the subneighborhood 6(x0) of Q(x”) containing x0 where 0 becomes injective, into the form represented by eqs. (1517) defined on @@(x0)) with:
system:
+ i+)+)
+ g(x)P(x)vll*l&3)
y =h(*l(z)); is of the form:
(14)
A =J(rl)
@ J(rz) 0 ...O
B =T(rl)
61 T(rz) 8 . . 0 T(rs);
(25)
C =S(rl)
8 S(rz) Q . . ‘8 S(rp);
(26)
(15)
i =f;(c,
(16)
rl);
Y =CC;
0 1
c E R’ x R”’ and the flinear sub09 system (A, B, C) is controllable as well as observable. The system represented by eqs. (15)(17) is said to be in the normal form.
(18)
B(x) =E(x;r) t ;
(19)
where Et is the pseudoinverse of E and Ljh= (L’ihr;.., L’ph,)*. If ]r] = ri +. . . + rp = n, define 0 : S2(x”) ) R” by
G =spen(gl,...,g,) (21) is involutive ( i.e. [gi,gJ] E B for every 1 5 i, j 5 m), choose n  ]r] solutions &l+r,. . . , #,, of L,/‘(X)
=o, &(x0) = 0
(22)
for all (r]t1 5 i 5 n, all 1 5 j 5 m and all x E Q(x”). Here 9 : Q(z”) ) R”, defined by
hi(x)  hl(x’) L;P‘h,(x)
 L;P‘hp(xo) 4lrl+l(x)
f&ix)
1 I
(23)
... ..’
0 0
0 1;TXt
0 T(r) =
0 0 1
; S(r)T = (l,O,...,O),, 1
and A @ B denotes the block matrix 4.3
A ( 0
0 B. >
Concept of Zero Dynamics
The design of a control system with a plant modelled by eq. (7) is reduced to the design of the corresponding linear subsystem (eqs. 15 and 17) and the analysis of the nonlinear flexible system (eq. 16) or, at least locally, to the study of the subsystem reprosented by i =f;(O,9), called the zerodynamics.
On the other hand, if ]r] < n and if the distribution
0 1 . ‘01 . :.
0 ( 0
where z =
o(x) =  E(x; r)tljh;
0
J(r) =
(17)
Let a system with the form in eq. (7) has a relative degree vector r at a state x0 E R”, with the associated maximal neighborhood Q(x’). Let:
(24)
where: 0
i =Ac + Bv;
J(rP);
11E P‘,
(27)
Indeed, if the linear subsystem is asymptotically stabilized and if the zerodynamics is locally asymptotically stable, then the entire closedloop system is also locally asymptotically stable. If instead the zerodynamics is unstable then the full system becomes unstabilizable. As mentioned earlier, the SISO version of the FLT can be derived readily by slightly manipulating the indices p and m (assigning the value 1 to each of them in the above expressions). This is shown in the application of the theory to the model of the flexible structure described in Section 2. 4.4
FLT as Applied to the Orbiting
MSS
By making use of the procedure outlined in Section 3, and by using what has been developed so far in this section, it is possible to implement effectively application of the FLT to the MSS. The objective is to effectively regulate the platform libration and provide joint trajectory tracking while avoiding obstacles.
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control strategy is outliied next.
Rrewriting the equations of motion as: Mf+F= Q, (28) where: x = {Es,, z:}* = {+,7~~,7~2,q*}* is the vector of generalized coordinates including the rigid and the flexible modes; fe,7 denotes the vector of the rigid modee to be controlled; and Z,, the vector of the flexible generalized coordinates. M, F, and Q have been defined earlier. Now, if only the rigid degrees of freedom are are controlled (one librational axis control moment and two sets of joint encoders), with high settling performance of the platform and joint tracking of the MSS, the procedure baaed on the previous analysis and generalized to account for the multiinput multioutput character of the system, gives the following normal form. Here the index is equal to the number of observed generalized coordinates corresponding to the rigid body dynamics:
j=l
A = G('D',
*‘VA) + kIfj(iI”,
(29) ‘Z’2,A)fj ;
j=l
y=
i&r’;
where @I = (+,7~1,7~2)*; q2 = 4’; A, the state vector of the internal dynamics; y, the output vector with components (I, 7~1, 752; and A, B, Q, P are matrices with appropriate dimensions involving \Ir’, 9’ and A. Furthermore, if the control effort Q is chosen as Q = A‘(@‘, ‘I”?,A)(B + v), the above equations can be expressed as: &I = 92;
W)
&2= v;
b = G’(*‘, \Tr2,A)+
&I;(@l, \lr2,A)v;
75
(31)
j=l
y= eir’; where v is the new designed input of the system; and G’, H’ arise due to transformation of G and H under the control Q. Using this procedure, the designer is able to completely decouple the rigid modes from each other and from the flexible ona. Note now two subsystems are constructed: one is completely controllable with the designer selected input while the other, describing the internal dynamic, of the system (unobservable) has to be at least critically stable. The remaining task for the designer is to combine this decoupling and settling effects of the rigid modes with a control strategy capable of assuring suppreeaion of the negative etkts of the vibrations created by the induced disturbances and the system flexibility. This
4.5 Active Vibration Suppression Following the action of the FLT, the system’s internal dynamics (the residual response resulting from the normal transformation) is still uncontrolled and exhibits harmonic oscillations. This is a side effect of the control procedure used here even if the acceleration and the velocity of all the rigid modes have been reduced virtually to zero. If not taken care of, the vibrations induced at the tip of the manipulator will persist indefmitly, unless some structural damping makes it decay to zero. This will lead to poor performance of the manipulator, especially if the tip tracking control is required. To deal with thii problem, an active vibration suppression scheme is proposed. It involves a set of colocated piezoelectric sensoractuator [email protected] , suitably positioned and activated towards the maneuver’s end. It efficiently damps the induced vibrations. This is motivated by the fact that near the end of the manipulator’s maneuver, the excitations induced by the rigid mode accelarations and velocities tend to zero, and the dynamics of the flexible modes is exclusively driven by the initial state of the system vibrations. This can be summarized as follows. Let the flexible dynamics be expressed as M=& + Fz, = f(+& + Qz,#  bn), WI where 2s denotes the vector associated with the flexible dynamics and QIp is a piezoelectric based actuator effort, activated towards the maneuver’s end (tm). Around tm, the system dynamics (flexible) can be approximated as linear, i = A(t)2 + B(t)u, (33) where: zT = (z:, i r ); and u(t) is the actuator effort. An optimal controllerobserver can now be designed to yield the dynamics i=Ai+Bu+L(yCCi), (34) where u, chosen optimally, is given by u = ki. Thii control scheme leads to an asymtotic decay of the state variables characterizing the flexible dynamics of the system. The combined action of the FLT with the active vibration suppression strategy described above leads to a composite control effort given by
Q _p = Q+i>vi + Q=q. (35) This control effort would generate, efficient settling and joint tracking performance while actively suppressing the induced disturbances. This is illustrated through the simulation examples presented next. 5
IIluetrative
Examples
It should be noted that the dynamical model as well as the composite control procedure adopted are
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reasonably genera1 and can be applied to a large class of planar systems with flexible Eulerian beam type links, elastic joints and a mobile base traversing a rigid platform in an arbitrary orbit. As the focus here is on the trajectory tracking in the presence of obstacles, and the joint as well as platform flexibiuy effects are well [email protected], the illustrative examples are so chosen as io help appreciate complex issues associated with collision avoidance. It considers a manipulator with two flexible links on a rigid platform negotiating a circular trajectory of 6000 s period, i.e. 200 km altitude. The base, nominally mobile, is purposely held fixed (near the tip of the platform) to further constrain manipulability, i.e. freedom in selecting the trajectory (Figure 2). The numerical data used in the analysis are summarized below: Space platform
(Frame Fp).
Length = 115.35 m; Mass = 240, 120 kg; Moment of Inertia, 1*,/I,, I,,/&
Fundamental htanipdatOr
I,,” = 2.67 x IO* Kgm’;
= l; = 0.0027;
(frames
Bending Frequency FJl,
= 0.2 Hz.
FJz)
To begin with, for assessing performance of the FLT baaed control , two simulations were carried out for the joint trajectory tracking with the gains assigned in such a manner as to produce critically damped tracking errors (with a natural frequency of 4 Hz) and underdamped tracking errora(with a natural fre quency of 5 Hz and a damping coefficient of 0.5, Figure 3). As can be expected, deviations from the ideal trajectory can be significant, particularly for the upper arm with an underdamped condition. It may be pointed out that while the total simulation time is three seconds, the maneuver is completed in two aeconds. Effectiveness of the active vibration suppression system was also evaluated for the critically damped situation as shown in Figure 4. At the outset, as expected, response of the second generalized coordinates for both the links and contribution of the first generalized coordinate for link 2 are rather small. On the other hand, 6~,,1 can be quite large (around 0.75m at 3 s). However, the active vibration control proce dure using piezoelectric elements successfully damps the upper arm vibration to an acceptable level. With this preliminary understanding of the system behavior, the attention was turned to the main task in hand, i.e. trajectory tracking in the presence of obstacles. Three different geometries of obstaclea were considered representing increasing order of difficulties in terms of collision avoidance:
Length of each Arm = 7.5 m; Mass of each Arm = 1800 kg;
(a) two obstacles;
Moment of Inertia,
(b) three obstacles;
I,,lI,,
= 1;
I,,/I,,
= 0.003;
Fundamental
I,, = 33750 Kgm2;
Bending Frequency = 0.16Hz.
The first two cantilevar beam modes are used for discretization of the arm deflections. The manipulator is located 50 m from the center of mass of the platform. By varying important parameters of the system in a planned manner, considerable amount of information was obtained. For conciseness, only a small sample of results useful in illustrating controlled performance of the manipulator is presented here. With the initial position of the manipulator specified and the final position fixed by the target, the ideal trajectory is first obtained by a cubic polynomial in conjunction with the constraints of zero initial and terminal velocity as well as acceleration. However, with the obstacles presenting possibilities of collisions, one may be forced to deviate from the ideal trajectory in accordance with the collision avoidance strategy as explained in Section 3.
(c) three obstacles in close proximity of the target. Furthermore, each of the case was explored using both a PD controller and the FLT. Results are presented in Figures 58. Performance plots for rigid links are also included to emphsize the influence of flexibility. Maneuver is the same as before, i.e. completed in 2 J, a demanding situation considering the fact that the link 1 slews through 30” (from the initial position of 90” to the final orientation of 60°, with respect to the horizontal). In fact the lower link slews through 60” in the same time! Figure 5 shows performance of the manipulator with the PD controller applied at the joints. For rigid links, the end effector reaches the target successfully, even in absence of damping, without collision. However, with the link flexibility accounted for, the situation is quite different. Now the manipulator tip strikes obstacle 2 as well s the target even with the level of damping ratio considered (1%). Note the deformations of link 1. Performance of the manipulator with the joint control based on the FLT and active vibration suppres
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00 Joint
2 \
Obstacle
Fixed base Target
Figure 2 Schematic representation of the MSS with obstacles used in the illustrative simulations.
Upper
 .desired
_.
0
1
(a)
arm
Time.s
2
3
Upper arm
(b)
Lower
..
0
1
Timr,s
2
3
arm
0.5
2.s
1
 auual
.
2
3
. _:;.*~ired
Tima,s
Figure 3 Joint tracking performancewith the FLT for: (a) the critically damped error dynamics; (b) the underdamped error dynamics.
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0.04
1
0.02
05
E ;
(2
E A 0 1.
‘.
0
2
0.5 I
$4.02
Ii. IA 0
1
Time.s
2
3
2
3
2
3
0.04~ 0
1
Time,s
2
3
0.04 0.02. 0 E. :_a02
YWY
co4) 1
olo
Times
;
Times
2
0.6 04
I
‘\ r
0.2
‘5
E f.
_ 0 W? 0.2 P
:i/
0.4 i 0
1
Time.s
0040
1
Time,s
2
3
Time.s
Figure 4 Time histories of the generalized coordinates with the FLT controlled regid degrees of freedom: (a) no vibration control; (b) active vibration suppression using five piezoelectric colocated sensors/actuators on each link.
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A sphere of influence
Figure 5 Performance of the manipulator under the conventional PD controller: (a) no damping; (b) with a passive damping ratio of 1% applied to each link.
_ sphere of influence
Figure 6 The FLT based control in the presence of active link vibration suppression through piezoelectric elements: (a) moderate damping ratio (2%); (b) larger damping ratio (5%) indicating higher penalty on the sphere of influence penetration.
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/
/
/
/
a 
/’
/
/
0
0
.
,
m
.
_
_.

._


_
FLT WITH STRONG ACTIVE DAMPING
Figure 7a Manipulator response showing several orientation (213 s apart) of the two links during a maneuver, in the presence of three obstacles, using four different control strategies
/
0
/
FLT WITH MODERATE ACTIVE DAMPING
target
CONTROLLER
CONVENTIONAL
flexible path
\ .
.
.
. rigid path
PD WITH PASSlVE DAMPING
\
\
.
.
l
l
FLT WITH MODERATE ACTIVE DAMPING
Figure 7b Manipulator response showing time histories of the end effector during the PD and FLT controls. Effect of link flexibility is also indicated.
PD ACTIVE
DAMPING
FLT WITH STRONG
f
z E

I_
.
.
target
.

_.
Fig. 8(a)
obstacle
sphere of influence
CONVENTIONAL PD CONTROLLER m / . . . _c/ 1’ I ,’

._
___

_
PD WITH PASSIVE DAMPING F ./ .* / / ,:’ / /
_
I_
.
0
./

/
.
.
_
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._
__

_
FLT WITH STRONG ACTIVE DAMPING
Figure 8a Response character showing typical orientations of the manipuktor (U3 s apart) for more critical placement of target with respect to three obstacles
.
,
F
FLT WITH MODERATE ACTIVE DAMPING
target
CONTROLLER
CONVENTIONAL
PO WITH PASSIVE DAMPING ACTIVE DAMPING
FLT WITH MODERATE
.
I’
.
.
.
FLT WITH STRONG ACTIVE DAMPING
trajectories for the manipulator considering Figure 8b Response character showing end effector links to be either rigid or flexible. Note, approach to the target is made more difficult by the deliberate placement of obstaclesas shown.
PD
6
5
h
4Sth IAF Congress
84
sion of the links is depicted in Figure 6. Two levels of piezoelectric induced damping levels were considered. For the rigid case, the performance is essentially comparable to that with the PD controller. However, the tip response shows improvement when the link flexibility is accounted for. As pointed out by the authors2p3, the FLT accounts for all nonlinearities of the system and provides complete decoupling between the rigid and flexible degrees of freedom, as well decoupling among the rigid generalized coordinates. Although, with a moderate level of active damping (2%), the lower arm does hit the second obstacle as well as the target, at a higher value (5%), impacts with obstacles as well as the target are entirely eliminated. Figure 7 considers a three obstacle situation. Four typical orientations of the links during the maneuver (2 s duration; 2/3 s apart orientations, i.e. showing initial, final and two intermidiate positions) as well trajectories of the endeffector are presented for both the PD and FLT schemes. The simulation is purposely continued for 3 s (1 s beyond the maneuver) to have some appreciation of the post maneuver response character. As in the previous case, the composite FLT/piezoelectric strategy with strong damping again proves to be successful. In fact, it retains the effectiveness even when the three obstacles surround the target at a closer range ( Figure 8), a challenging situation. 6
Concluding
Remarks
An obstacle avoidance procedure for a Mobile Flexible Manipulator (MFM), supported by a flexible platform, is developed which appears to be quite promising. It is based on a composite control scheme with path planning accounting for spheres of influence around specified fixed obstacles. Application of the algorithm is illustrated through several examples, involving two or three obstacles. Relative merit of the PD control against the composite FLT with active vibration suppression is assessed. The influence of damping level is also studied in each case. Results suggest that the composite FLT control with an appropriate level of damping leads to the best performance while preserving the highest level of manipulability. Even with three obstacles surrounding the target at close proximity, the end effector is able to reach the destination without collision. Ruthermore, the maneuver itself is rather quick as the 30° slew of the upper arm is completed in only 2 s. Thus the control algorithm has proved to be effective under demanding situations. In the present study, joint trajectories were considered to be available in real time. This of course, is a challenging task. In fact, instantenous imple
mentation of the control algorithm is neither possible In practical applications, depending nor necessary. upon the task, certain level of delay is usually acceptable. Thus locations of the obstacles are sensed, desired end effector’s trajectory established, and then the inverse kinematics is used to provide the actuator joints with the commanded trajectory. Obviously, this can be better accomplished through a supervisory control procedure involving parallel processing of an integrated computer vision system with path planning in conjunction with a low level composite control scheme. It represents natural extension to the current study. 7
Ill
References
“Space Engineering Data Book,” NASA SSEENASA Space Station Program Office, USA, November 1987. 87RI,
I21Karray,
F., Modi, V.J., and Chan J.,“On The Nonlinear Dynamics and Control of Flexible Orbiting Manipulators,” Journal of the Nonlinear Dynamics, Vol. 5, 1994, pp. 7191.
[31 Modi, V.J., Karray, F., and Mah, H., “Composite Control Scheme with Vibration Suppression for a Class of Manipulators with Flexible Links,” 44th International Astronautical Congress, Graz, Austria, October 1993, paper No. IAF 93A.4.29; also Acta Astronautica, in press. I41 Nakamura, Y., Hanafusa, H., and Yoshikawa, T.“TaskPriority Based Redundancy Control of Robot Manipulators,” International Journal of Robotics Research, Vol. 6, 1987, pp. 315. PI Chan, J.K., and Modi, V. J., “ Dynamics and Control of an Orbiting Flexible Mobile Manipulator”, Acta Astronautica, Vol. 21, No. 11/12, 1990, pp. 759776. PI Chan, J.,“Dynarnics and Control of an Orbiting Space Platform Based Mobile Flexible Manipulator,” M. A. SC. Thesis, Department of Mechanical Engineering, University of British Columbia, April 1990. 171A. Isidorl, Nonlinear Control Systems Springer Verlag, BerlinHeidelburg, 1989, pp. 144230 . PI Murotsu, Y., Tsujio, S., Senda, K., and Hayashi, M.,“Trajectory Control of Flexible Manipulators on a Flying Space Robot,” Control Systems Magazine, Vol. 12, No. 3, June 1992, pp. 5157. PI Devasia, S., Mereasi, T., and Paden, B., ” PiezoElectric Actuator Design for Vibration Suppreasion: Placement and Sizing,” Proc. of the 31 st IEEE Conference on Decision and Control, Arizona, December 1992, Vol. 2, pp. 13671372.