IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 11, NO. 6, DECEMBER 2006 699
Autonomous Robotic Capture of a Satellite UsingConstrained Predictive Control
Richard A. McCourt and Clarence W. de Silva, Fellow, IEEE
Abstract—This paper investigates the use of model-based predic-tive control for the capture of a multi-degrees-of-freedom objectthat moves in a somewhat predictable manner, using a deployablemanipulator. The study is conducted through both computer sim-ulations and ground-based experiments with the intended applica-tion focused on automating the robotic capture of a free-floatingand spinning satellite. The present investigation uses an innovativemanipulator known as the multi module deployable manipulatorsystem (MDMS), which has been designed and built in our labo-ratory. The MDMS is used to capture the simulated target satel-lites. The objective is to evaluate the performance of the predictivecontroller for the capturing task and to investigate the need andfeasibility of incorporating constraints into the controller. Offlinesolutions to multi parametric quadratic programming (mp-QP)problems are used to create a lookup table so that constrainedoptimal control decisions can be made in real time. Moreover, asuboptimal formulation of the mp-QP problem has been used toreduce the size of the online lookup table and to mitigate any nu-merical problems. The results show that when the satellite motionis predicted, the tracking performance is improved. Furthermore,when the physical constraints of the system are formulated into theoptimization, the controller becomes aware of its own limitationsand the approach toward the satellite is improved by eliminatingthe possible overshooting of the target.
ONE of the many reasons for the development of roboticmanipulators is the ability for them to perform tasks that
are not particularly suitable for humans; for example, those in-volving long and repetitive operations, and unhealthy, unpleas-ant, and hazardous environments. Due to the particularly harshenvironment of space, the application of robotics has receivedsignificant attention. An important issue concerning the pres-ence of humans in space is the servicing of the large numberof satellites that orbit Earth. A special-purpose satellite that hasreceived a considerable amount of attention is the Hubble spacetelescope (HST). In the past, astronauts, with the help of theCanadian robot Canadarm, have been sent on board the spaceshuttle to perform various repairs on Hubble telescope in orderto keep it operational. However, recent efforts to repair the tele-scope have been rethought due to increased safety concerns [1].
Manuscript received December 3, 2004; revised June 20, 2006. Recom-mended by Technical Editor M. Meng. This work was supported in part by theU.S. Department of Commerce under Grant BS123456.
R. A. McCourt is with the Department of National Defence, Ottawa, ON K1A0K2, Canada (e-mail: [email protected]).
C. W. de Silva is with the Mechanical Engineering Department, Uni-versity of British Columbia, Vancouver, BC V6T 1Z4, Canada (e-mail:[email protected]).
Digital Object Identifier 10.1109/TMECH.2006.886246
Efforts to save the space telescope have led to heightened inter-ests in the area of robotic satellite servicing, with the emphasison telerobotics.
The successful robotic servicing of the Hubble telescope willundoubtedly give a further boost to the growing interest in un-manned orbital service vehicles (OSVs). These OSVs would becapable of remaining in orbit for very long periods of time andwould be sufficiently versatile to service a large variety of orbit-ing space traffic. MacDonald, Dettwiler and Associates (MDA)and Boeing Phantom Works are doing some research in this areawith their “orbital express” program. Other examples of OSVsare found at the National Space Development Agency of Japan(NASDA) with their hyper-OSV [2], and the Robotic SystemsTechnology Branch of NASA with their tele-operated android,named Robonaut [3], [4].
The control of these OSVs would typically be achievedthrough human tele presence control systems, allowing it tobe operated safely from a ground control station on Earth. Un-fortunately, the communication delay between the OSV and theground control station can be of the order of several seconds.This delay would add to the difficulty that a remote operatorwould face in maneuvering a manipulator in a complicated task,not the least of which is system instability. One task that hasproven to be particularly difficult in the past is the maneuveringof a robot in the capture of a moving target. Several satelliteshave been captured for servicing or repair with the help of theCanadarm robotic manipulator on the space shuttle; however,the task can become extremely difficult when the satellite is spin-ning or tumbling out of control. For example, during STS-87,an astronaut operating the Canadarm attempted to capture a freefloating SPARTAN satellite. In this attempt, the end-effector ofthe manipulator nudged the target satellite, causing it to tumble,thus making it more difficult to complete the capture, therebysending the astronauts out on a space walk to capture the satel-lite by hand. Of course, in the case of an unmanned OSV, humanaid will not be utilized.
Automating the satellite capturing task would reduce the risksassociated with these operations and the mission costs, increasethe likelihood of successful capture, and in general improve theeffectiveness of the OSV. Some research has been done in thearea of automated satellite capturing task, [5]–[9], but so far,the only autonomous satellite capture performed while in orbithas been done on the NASA’s engineering test satellite (ETS)VII [10].
If the future of space robotics rests on the tele-operated andautonomous systems, then autonomous moving target capturingshould be included in the repertoire of a manipulator’s abilities.One such manipulator that possesses several advantages over
700 IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 11, NO. 6, DECEMBER 2006
the standard all revolute joint space manipulators is the multi-module deployable manipulator system (MDMS) as conceivedin our laboratory. This paper focuses on the use of the model-based predictive control for this robotic capturing task.
Predictive control has been applied to a variety of mobilerobotic applications and a few robotic manipulators. For thelatter, the controllers have the added difficulty in dealing withthe nonlinearities of the manipulator. Some robotic predictivecontrollers first linearize the robot using the feedback of theinverse dynamics of the manipulator [11], [12], while othershave used Taylor series approximations [13] or linearization ofthe robot at each time step [14].
In the aforementioned references, a good simulation resultsappear to have generated when predictive control was appliedto a robot, but such claims have not been validated, for the mostpart, through experimental investigation and practical applica-tion. This paper integrates such efforts in predictive control ofrobotic satellite capturing with the MDMS, using experimentalimplementation and verification as well as analysis and com-puter simulation. First the description of the MDMS and thepredictive controller are provided, and then the benefits of us-ing predictive control are demonstrated through both computersimulations and experimentations on a ground-based prototypeof the manipulator that has been developed in our laboratory.
II. MDMS
The MDMS is a robotic manipulator comprised of a series oftwo-degree-of-freedom (DOF) modules connected together in achain topology. Each module is a combination of one slewing(revolute) link and one deploying (prismatic) link. The manipu-lator may be configured for a particular application by connect-ing as many modules as necessary. The MDMS has a variablegeometry structure, in view of its deploying links, which pro-vides particular advantages in obstacle avoidance. It has beendesigned as a multi-purpose manipulator, and the satellite cap-ture problem is investigated here as a specific application of thisclass of manipulators. It is similar to the mobile servicing system(MSS) on the International Space Station; however, compared toconventional (all revolute joint) manipulators, like the MSS, theMDMS has several advantages. In addition to better obstacleavoidance, the MDMS has fewer singular configurations, re-duced dynamic coupling between the links, and simpler inversekinematics.
In this paper, a ground-based prototype of the MDMS, asdeveloped in our laboratory, is used in the experimental inves-tigations. The physical prototype is a planar manipulator withrolling supports under the manipulator joints, which is used toeliminate the gravity effects. The MDMS prototype, in a con-figuration with two modules, is shown in Fig. 1. The base ofthe manipulator is located at the top right corner of the figure,and the first module is fully extended to display the prismaticjoint. The revolute joints of the manipulator use dc motors fittedwith low backlash, harmonic drive gears, and deployment inthe prismatic joints is provided by dc motors directly coupledto ballscrew actuators. Feedback motion signals from the jointsare sensed using incremental optical encoders attached to the
Fig. 1. MDMS prototype used in the experiments.
Fig. 2. Schematic diagram of the MDMS used for planar satellite capturing.
motor shafts, and are fed to a servo card installed on the robotcontrol computer running Windows 2000. The controller is pro-grammed in C++, and VenturCom’s real-time extension (RTX)provides the application programming interface (API), therebygiving real-time capabilities to Windows [15], [16]. An illustra-tion of the manipulator in a similar configuration is shown inFig. 2, and the key parameters used in the model implementationhave been labeled.
A. Dynamics of the Manipulator System
Even though the MDMS has been designed to possess kine-matic redundancy, this capability is not considered. Specifically,to reach the desired position and orientation of the target inplanar motion, only three joints (three (DOF)) are needed. Ac-cordingly, only two modules of the MDMS are chosen, and theprismatic joint of the distal module is not being used in the exper-imental investigation. A schematic diagram of the manipulatoras used in this paper is shown in Fig. 2.
The manipulator in Fig. 2 is assumed to be fully rigid, specifi-cally with rigid links and rigid joints. A space-based manipulator
MCCOURT AND SILVA: AUTONOMOUS ROBOTIC CAPTURE OF A SATELLITE 701
may possess both joint and link flexibility, and in that case,the corresponding model should include these effects [17].Fortunately, the prototype MDMS is known to be quite rigid,and the flexibility effects can be neglected for most practicalpurposes. In the figure, mi and Ii represent the mass and secondmoment of area, respectively, of the ith link. θ1 is the slew angleof joint 1 (a revolute joint), d2 is the deploy length of joint 2(a prismatic joint), and θ3 is the slew angle of joint 3 (a revolutejoint). ls is the length of the slewing link and ld is the lengthof the deploying link. In the case of the most distal link, ls hasalso been defined as lw, which is the length of the “wrist” link.
The flexibility and redundancy of the manipulator are not theonly characteristics of the space-based system that are differentfrom the ground-based system. In the space-based system thebase of the manipulator is fixed to a free-floating platform,hence, the movement of the manipulator will affect the positionof the target relative to the base of the robot. In this paper, it isassumed that:
1) constraints can be placed on the operational speed of themanipulator to limit the forces at the base of the robot;
2) the mass of the platform base is sufficiently large, so thatany forces generated at the base of the robot will result ina negligible movement of the base relative to the target.
The dynamics of the robotic system, as shown in Fig. 2, maybe determined using Lagrange equations, and then static andviscous friction terms may be added as generalized terms togive the vector-matrix equation
M(q)q + C(q, q) + F ssgn(q) + F vq = K u (1)
whereq, q, q vectors of joint positions, velocities, and accelera-
tions, respectively;M(q) inertia matrix;C(q, q) vector containing centripetal and Coriolis terms;F s static friction terms;F v viscous friction terms;K conversion factor from servo amplifier command
voltage to torque/force at joint;u vector of command voltages computed from the
controller.The absence of any gravity effects in (1) is due to the micro-
gravity operating environment of the space-based system and thehorizontal plane of operation of the ground-based manipulator.
B. Linearizing the MDMS
The predictive controller, discussed in this paper, uses a lin-ear model of the plant in the optimization of the cost function.However, robotic manipulators are globally nonlinear plants;therefore, a linearization of the MDMS must be made beforethe controller can be used. This may be achieved locally us-ing Taylor series expansion about an operating point, or moreappropriately, using global linearization through feedback lin-earization, where the inverse dynamics of the robot are appliedin a feedback loop around the system [18]. This global lin-earization is possible in view of the availability of a completenonlinear model of the MDMS.
Alternatively, a physical approach to linearizing a robot ma-nipulator is to fit the joint motors with gear reducers havinglarge speed reduction. In this manner, the joint torques/forcescan be highly amplified and will form the inputs to the plant. Theoutputs will be the joint positions and velocities, and the non-linear effects are left as unmeasured disturbances, which maybe considered small when compared to the amplified inputs.Some of the disadvantages of placing large gear ratios on thejoints are that the gear reduction greatly decreases the maximumspeed of the joint, and the accuracy of the manipulator can belost due to added nonlinearities in the gears, such as backlashand friction. Fortunately, however, fast manipulation speeds arenot needed and are undesirable for space manipulators becauseof their long, flexible structure and the sensitivity of the equip-ment that is being manipulated. In addition, the ground-basedprototype of the MDMS has been designed with harmonic geardrives, which virtually eliminate backlash and have very highgear ratios [19], [20]. For these reasons the predictive controllerwas implemented as a direct, low-level controller on the MDMS.
The joint positions and velocities are combined into a statevector, x = [qT qT ]T ∈ �n, and the system dynamics in (1)may be expressed in the state-space form as
[qq
]=
[q
−M−1(q){C(q, q) + Fssgnq + F vq}
]
+[
0M−1(q)
]K · u. (2)
The model is first linearized about an operating configurationwith the prismatic joint fully retracted and the distal module atan angle of zero, as defined in Fig. 2, i.e., q = [0 ls 0]T andq = [0 0 0]T . Then, the model is discretized with a samplingtime Ts of 5 ms, which is large enough for the optimal solutionto be found within the sampling period without compromisingthe performance of the controller. The linearization and dis-cretization are done using the “dlinmod” function in MATLABand ls = 0.405 m to obtain
where x(k + 1) is the one-step predicted state vector, madefrom the current state vector x(k) and the current control inputu(k).
702 IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 11, NO. 6, DECEMBER 2006
Fig. 3. Sequence of motion of a spinning satellite.
III. TARGET KINEMATICS
In general, the controller developed in this paper may beused for robotic tracking of any moving target subject to thekinematic and dynamic limitations and the operating bandwidthof the manipulator. However, in the simulations and experimentsperformed, the target is assumed to be a free-floating satellite,i.e., passing by in the absence of external forces or torques.
As mentioned earlier, the present mode of operation of theMDMS possesses three-(DOF) in planar motion. This is be-cause, unlike catching a ball, a satellite typically has a safegrappling point where the manipulator must try and grab it,and it is desirable for the manipulator to have both the cor-rect position and orientation when the capture is being exe-cuted. Only planar models are considered so that experimentscan be completed using the planar robotic manipulator in ourlaboratory.
The automated capturing task is implemented to alleviatesome of the difficulties encountered when capturing a spinningsatellite. The grappling point on the satellite is not moving ina straight line, but rather spinning around the moving centerof mass of the target, as shown in Fig. 3. Here, (xc, yc) is thelocation of the target center of mass, which is rotating in theplane of the page with angular an velocity ω. The resultanttarget position is given by coordinates (xt, yt).
One of the many advantages of the MDMS is its simpleinverse kinematics solution. For the nonredundant manipulator,as shown in Fig. 2, the inverse kinematics solution is uniquelygiven by
θ′1 = a tan 2 (ye − lw sin θe, xe − lw cos θe)
d′2 =√
(xe − lw cos θe)2 + (ye − lw sin θe)2
θ′3 = θe − θ1 (4)
where (xe, ye, θe) describes the end-effector position and orien-tation, and lw is the length from the end-effector to the center forthe third joint. Equation (4) can be used to express the positionand orientation of the grappling point on the satellite in the jointspace of the MDMS.
The target point (xt, yt) on the satellite is located at a dis-tance lt from the center of mass of the satellite at (xc, yc), and isoriented at the angle θt. The center of mass is moving at a con-stant velocity of v = [vx vy]T , and spinning with a constantangular velocity of ω. As a result, the position of the grappling
point can be expressed as
xt = x0 + vxt − lt cos(θ0 + ωt)
yt = y0 + vyt − lt sin(θ0 + ωt)
θt = θ0 + ωt (5)
where (x0, y0, θ0) is the position and orientation of the cen-ter of mass at t = 0. The substitution of (5) into (4) will givethe joint-space position of the target, and the result is formedinto a vector of joint references r(k) = [θ′1(k)d′2(k)θ′3(k)]T ]for the controller. r(k) is the desired joint-space position ofthe manipulator. The substitution of (5) into (4) incorporatesthe nonlinearities of the robotic manipulator and the spinningtarget. Therefore, this joint-space representation of the target isapproximated with a second-order Taylor series expansion inthe implementation of the controller.
IV. PREDICTIVE CONTROLLER
The autonomous satellite capturing controller must be able tosmoothly maneuver the manipulator toward an expected inter-ception point with the target and continue to track the target untilthe capture is complete. Since the target will be moving withincompletely known motions, it is desirable to have a controllerthat is capable of anticipating the future movement of the target.In addition, if the controller is aware of its own capabilities,it can use this knowledge and any available knowledge of thetarget to help in choosing the best robotic action.
Model predictive control (MPC) is a model-based optimalcontroller, which uses open loop predictions of the system re-sponse into a finite, future length of time to repeatedly minimizea user-defined cost function in achieving the specific control ob-jective. Predictions can be made on both the target and the robotto create the desired anticipative control action. The specificpredictive controller used in this paper has the ability to handlethe user-defined constraints. These constraints can be used tospecify the desired level of smoothness in the joint motions.For these reasons MPC is particularly suitable for use in theautonomous satellite capturing robots.
The controller is constructed assuming that the expected joint-space position of the target is known. As mentioned earlier,the expected joint-space location of the target has been chosenas the reference value. Typically, in robot motion control, atrajectory is generated to specify how the controller should movethe manipulator between any two points. However, since theproblem addressed in this paper is a robotic capturing task, the
MCCOURT AND SILVA: AUTONOMOUS ROBOTIC CAPTURE OF A SATELLITE 703
robot must somehow span the distance between some initialstarting position and the desired position of the target before itis able to follow the reference point like a standard trajectorytracking problem.
Some work has been done in the area of online trajectoryplanning for robot catching, for example, [21] and [22]. Thecontroller developed in this paper directly minimizes the jointmotion differences between the reference target and the currentstate of the manipulator. It does this by trying to anticipate themotions of both the target and the manipulator. The controlleris tuned to provide the desired tracking performance, and theoutput constraints are used to control the speed at which thejoints approach their target values.
MPC has been successfully used in the process control in-dustry [23]; however, the application of predictive control inthe aerospace industry appears to be relatively new. The recentapplications of predictive control in high bandwidth aerospacesystems have been aided by work in muti parametric, explicitMPC solutions [24], and with the constantly increasing speedof computers.
A. Controller Formulation
In this paper, a quadratic cost function is used. The costfunction minimized in the tracking control problem is
V (k) =N∑
i=1
‖y(k + i|k) − r(k + i|k)‖2Q(i)
+N∑
i=1
‖∆u(k + i − 1|k)‖2R(i) (6)
where ‖y(k + i|k) − r(k + i|k)‖2Q(i) is the two-norm of the
joint-space tracking error between the manipulator y(k + i|k)and the target satellite r(k + i|k) at the future time t = (k +i)Ts weighted by the diagonal and the positive semi-definiteweighting matrix Q(i) ≥ 0. The future changes in control inputs∆u(k = i|k) are penalized in the cost function as a means ofproviding offset free tracking, and the amount at which they arepenalized at time t = k + i − 1 is weighted by the diagonal andthe positive definite matrix R(i) � 0.
A basis function parameterization of the control inputs hasbeen chosen for this controller. A first-order parameterizationwas chosen, so the current and future input vectors are relatedthrough the polynomial function
∆u(k + i|k) = µ0(k) + µ1(k)i (7)
and the vectors µ0(k) and µ1(k) become the argument of theoriginal minimization problem in (6). The basis function param-eterization of MPC is known as predictive functional control(PFC) and was originally developed by Richalet et al. [25], [26]in the early 1980s for application of MPC on high-bandwidthservomechanisms. PFC has several advantages over the controlhorizon form of the controller. In general, PFC is capable ofmaking more accurate predictions for a better-fit solution to theminimization problem in (6), especially when tracking com-plex trajectories [27]. In the unconstrained controller, (6) can
be solved explicitly for the optimal control input change ∆u.However, for reasons mentioned previously, constraints havebeen considered, and constrained optimization methods mustbe used when minimizing (6).
B. Constraints
The controller must take the manipulator from its initial con-figuration to an interception point, and then continue to trackthe target. The tracking may be aborted if the target moves outof range of operation. It is desirable to have the controller tunedso that the joint space tracking error is minimized during thetime the target is within the operating space of the manipulator.However, when the controller is tuned to give the desired track-ing performance, the initial large reference errors lead to largeinputs torques. Subsequently, as the manipulator approaches thetarget, large joint velocities are generated. These are undesirableeffects because the target is a satellite floating in a microgravityenvironment, and any impact forces could possibly be damag-ing, and would cause the target path to change directions andpossibly float away out of control.
Constraints are added to control how smoothly the manip-ulator approaches the target. Since the plant is linearized, in-put constraints can be used to put limits on the joint accel-erations, and constraints on the change in the input can pro-vide jerk limits. The joint velocities and positions are outputsof the controller, and using the linear model of the MDMS,(3), all of these constraints can be formulated in terms ofthe control input, i.e., x(k + 1) = f(x(k),u(k)). Addition-ally, the control input u(k) can be determined from the pre-vious input value u(k − 1), and the basis function parametersof the change in control input µ(k) = [µ0(k)T µ1(k)T ]T de-fined in (7). The result is x(k + 1) = f(x(k),u(k − 1),µ(k)).Now, for any constraints on the future states of the manip-ulator, i.e., x(k + 1) ≤ W , for some constant constraint W ,and using the linear model in (3), the constraints can bewritten as
Gµ(k) ≤ W + E[
x(k)u(k − 1)
](8)
where G, W , and E are the matrices used to describe theconstraints on the system in terms of the coefficients in µ(k) =[µ0(k)T µ1(k)T ]T . For example, the constraint on the ith statexi can be represented by adding (3) and (7)
xi(k + 1) = Aix(k) + Biu(k)
= Aix(k) + Bi(u(k − 1) + µ0(k) + µ1(k))
= [Ai Bi ][
x(k)u(k − 1)
]+ Biµ(k) ≤ Wi
where Ai and Bi are the ith rows of the matrices premultiplyingx(k) and u(k), respectively, in (3).
The controller is implemented in joint space, as a physical ne-cessity, but it is the task space maximum approach velocities thatare important. However, the task space velocities are nonlinearlyrelated to the joint space velocities through the robot kinematics,and constraints on the maximum task space velocity would lead
704 IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 11, NO. 6, DECEMBER 2006
to nonlinear constraints in joint space. Nonlinear constraints arenot considered in this paper, and consequently the constraintson the joint velocities only provide a rather conservative upperbound on the task space velocity.
In the absence of constraints, the optimal minimization of (6)is a function of the current state, the previous input, and the de-sired setpoints and a feedback matrix can be computed offlinein order to achieve fast implementation online. However, ingeneral, the constrained minimization of (6) can be highly com-putational intensive. Therefore, methods for explicitly solvingthe multi-parametric quadratic programming (mp-QP) problemoffline have to be considered. Such a scheme has been developedby Bemporad et al. [24] and is discussed next.
C. mp-QP
In the minimization of the cost function of the predictivecontroller, the parameters that are changing at each time stepare the measured states of the robotic manipulator x(k), theprevious control inputs to the plant u(k − 1) and the state ofthe desired values r(k). The values of these parameters willdetermine the location of the unconstrained minimum point inthe space defined by the optimization variables µ(k) ∈ �Mnu ,and the location of this minimum point will correspond to aspecific set of active constraints. In mp-QP these regions ofconstant active sets are determined offline, and using the first-order Karush–Kuhn–Tucker (KKT) conditions, and an affinesolution is determined for each of these regions. The solutionis piecewise affine in a space defined by the Np parametersthat change in the optimization at each time step; i.e., the nstates, the nu previous inputs, and the nr current and predictedreferences. The parameter space is partitioned into a total ofNr regions and these regions form a lookup table that can beused online for fast implementation of the constrained solu-tion. The multi-parametric toolbox (MPT) [28] of MATLABis used to find the Nr regions and the corresponding affinesolutions. Further details on multi-parametric optimization tech-niques can be found in [24].
Searching for the current active region is done via a sequentialsearch of all the regions. If the search runs through all theregions and is unsuccessful in finding the appropriate controllaw, then the controller reverts to the unconstrained solution,but with much more restricting saturation values. This methodis chosen because it ensures that the manipulator continues tomove toward the target, however, it chooses more conservativemotion toward the target until a feasible parameter value can befound.
D. Integration of the Predictive Controller
All things being equal, the mp-QP optimization should pro-duce exactly identical results as that from online computation.Even though this could be true, the number of optimizationparameters Np in the MDMS satellite capturing problem canlead to incredibly large lookup tables. In addition to the prob-lem of large lookup tables, it was found that the mp-QP solverin the MPT was much more sensitive to ill-conditioned prob-lems than the QP solver that was used at each time step in the
previous simulations. The numerical difficulties arose when theconstraints were defined at every future time in the predictionhorizon. Enforcing the constraints only at the selected pointsthroughout the prediction horizon often mitigated the numericalproblems, and reduced the size of the lookup table; however,this also made the controller suboptimal. Even so, the subopti-mal mp-QP controller was found to be better than the uncon-strained controller because it had some knowledge of the inputconstraints and was was able to realize the user defined outputconstraints.
In the constrained controller, the vectors x(k), u(k − 1),and r(k + i|k) for i = {1, 2, . . . , N} are the Np parameters ofthe optimization, and in the mp-QP algorithm they form theparameter space that is partitioned in the offline optimization.In order to further decrease the size of the parameter space andincrease the computational efficiency, a second-order Taylorseries approximation of the target model is formulated into theproblem. We get
Here r (k) and r (k) are vectors of the current joint referencevelocities and accelerations, respectively, and Ts is the samplingtime of the controller. Equation (9) assumes that the current jointacceleration is constant throughout the prediction horizon. Sincethe window of time in which the controller looks into the futureis small relative to the dynamics of the target, the approximationgiven by (9) is adequate.
When the joint velocities and accelerations are added to theproblem, care must be taken when passing near, or through, sin-gular configurations where the velocities and accelerations canbecome very large. Fortunately, there is particular advantage ofthe MDMS in this regard, as it has fewer singular configurationsthan a standard, all-revolute joint manipulator. For the setup ofthe MDMS, as shown in Fig. 2, there are no singularities in theoperating space of the manipulator. The only singularity occurswhen joints 1 and 3 occupy the same space, but the limits ofthe prismatic joint prevent the robot from ever reaching thatconfiguration.
A block diagram of the controller developed is shownin Fig. 4. In the diagram, the vectors x(k), u(k − 1), andr(k + i|k) are the parameters of the affine function of the op-timal solution. They are multiplied by the gain F i, and off-set by the term Gi. F i and Gi are the terms returned bythe lookup table that was produced offline from the mp-QPsolver.
V. SIMULATION STUDY
In the simulations, the manipulator starts from rest, lying par-allel to its platform base, and the MDMS attempts to catch asatellite that is moving with velocity v = [vx vy]T and spin-ning with an angular velocity ω, as shown in Fig. 5.
The model parameters used in the design of the controllerare found in [20]. The parameters defined in Fig. 2 are given inTable I.
MCCOURT AND SILVA: AUTONOMOUS ROBOTIC CAPTURE OF A SATELLITE 705
Fig. 4. Block diagram of the satellite capturing controller.
Fig. 5. Simulation model of robotic satellite capture.
TABLE IPARAMETERS OF THE SIMULATED MDMS
A. Unconstrained Predictive Control
The optimal solution to the predictive controller is first com-puted in the absence of constraints. Since in a real systemthere are always constraints, when the unconstrained controlleris implemented the command signals computed by the con-troller are saturated at the maximum output of the servo card(±10V).
Several different target cases are used. As an example, con-sider a target moving parallel to the platform with a veloc-ity v = [0 −0.05] m/s and rotating with an angular velocityω = π/60 rad/s. The simulated target point is initially 0.9 metersaway from the end-effector with an initial misalignment of75◦. When a prediction horizon of 30 and the second-orderapproximation of the target are used, the tracking error isreduced to less than 5× 10−4 m and 0.5◦ in approximately
2 s. However, in achieving these results, the manipulator jointsovershoot their target positions. The only way this can becorrected in the unconstrained controller is by decreasing theweights on the position tracking errors relative to the weightson the velocity tracking errors. Unfortunately, when less impor-tance is placed on the joint position errors the tracking perfor-mance suffers.
When the saturation of the control input signal is removed, thedegree of overshoot is significantly reduced; however, very largecontrol inputs are needed. The decrease in the overshoot is due tothe fact that the controller is making more accurate predictions.The controller is able to apply the calculated optimal solutionwithout the inputs being saturated. So, when the controller isunaware of the abilities of the plant (i.e., its input and outputconstraints), the minimization of the plant error cannot be fullyrealized, and the performance degrades.
B. Constrained Predictive Control
When input constraints are used, the controller makes moreknowledgeable decisions. To demonstrate the ability of the mp-QP controller, constraints were placed on the inputs only atthe beginning and the end of the horizon. The result was alookup table with Nr = 384 regions and took just over 5 minto compute on a Pentium4 2.4 GHz processor in MATLAB 6.1.This set of constraints was chosen because when the constraintswere placed on the inputs over the entire horizon, the size ofthe lookup table became very large and the mp-QP solver raninto numerical problems. With this set of constraints defined,the controller predicted that it was able to move slightly beyondthe constraint in the middle of the horizon, but because theoptimization was performed at each time step and the maximuminput constraints were satisfied at the beginning of the horizon,no constraint violations occurred.
The results from the simulation with the mp-QP controller areshown in Fig. 6. The results show that despite the suboptimalpredictions, the controller is still able to intercept the target with-out overshooting the target values. The unconstrained solutionis also plotted in Fig. 6 to show the benefit of incorporatingconstraints into the control decision.
As a means of comparison and to show the full benefitsof constrained predictive control, the optimal solution is alsocomputed by solving the optimal control problem at each timestep using the “quadprog” QP solver from the optimizationtoolbox in MATLAB. These results are superimposed in Fig.6 to compare the performance of the fully constrained andthe suboptimally constrained controllers. The results from themp-QP controller are more aggressive because the controlleris not fully aware of the constraints that will be met when themanipulator has to decelerate upon arrival at the target. Once theapproach toward the target is complete and the MDMS is track-ing the target, none of the constraints are active and the mp-QPsolution produces the same results as the QP and unconstrainedsolutions. The tracking error is reduced to less than 1 mmand 0.5◦. So, other than the slightly more aggressive approachtoward the target, the suboptimally constrained mp-QP solution
706 IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 11, NO. 6, DECEMBER 2006
Fig. 6. Simulation results.
and the fully constrained QP solution produce very similarresults.
VI. EXPERIMENTAL INVESTIGATION
One of the main advantages of RTX for Windows is theease of use. This was certainly true for the predictive controllerdeveloped here. With RTX, the controller can be run on thesame computer immediately after the mp-QP solution has beencomputed in MATLAB and, therefore, making the developmentprocess much easier.
The MDMS used in the experiments does not have an end-effector capable of capturing targets. Consequently, for the ex-periments carried out here a simulated target has been used. Thecoordinate system for the experiments has been kept the sameas shown in Fig. 5.
When the unconstrained predictive controller was applied tothe prototype manipulator [29], the experimental results werefound to be slightly more damped than the corresponding simu-lation results and the end-effector did not track as closely to thetarget as desired. In view of this, when running the subsequenttests on the manipulator the weighting matrices were adjusted togive the desired performance. The results from target satellites
TABLE IIEXPERIMENTAL UNCONSTRAINED CONTROLLER RESULTS FOR VARIOUS
CONDITIONS OF THE TARGET SATELLITE
Fig. 7. Experimental results.
at different velocities are summarized in Table II. Positional ac-curacy is given as the total separation distance, while orientationaccuracy is given as the rotational tracking error, i.e., negativevalues mean the end-effector orientation is leading the targetorientation.
In the tests with the unconstrained predictive controller over-shooting the target value did not occur as much as in the sim-ulations; however the manipulator did approach the target withundesirable (fast) joint velocities. The third joint generated thehighest velocity of approximately 3.5 rad/s, and in Case 2 thisjoint overshot its target value. Consequently, the first outputconstraint was defined so that the joint velocity of the third joint
MCCOURT AND SILVA: AUTONOMOUS ROBOTIC CAPTURE OF A SATELLITE 707
remained below 0.75 rad/s. This value was chosen so that thejoint would still be able to track its target without violating theconstraint. The next constraint was placed on the joint velocityof the prismatic joint (joint 2). In the unconstrained controllerthe inputs to this joint were saturated during its approach to-ward the target, which resulted in a maximum joint velocity ofapproximately 0.225 m/s. A constraint was placed for the jointvelocity to remain below 0.1 m/s, which was, again, chosenso that it would not limit the ability of the joint to track itstarget.
The joint constraints were defined only at the beginning andthe end of the prediction horizon, which resulted in a controllerconsisting of Nr = 92 regions. A controller sampling frequencyof 200 Hz was used, and the sequential search was able to searchthrough these 92 regions in an average of 0.5 ms on the Pentium42.4 GHz PC used in the experiments.
The results from Case 4 with the unconstrained and con-strained predictive controllers are given in Fig. 7. These resultsshow the constrained joint velocities in the second and thirdjoints, but more importantly they show an improvement in theway the manipulator approaches the target and the inputs gen-erated by the controller. When the velocity of the third joint isconstrained to 0.75 rad/s the level of overshoot is eliminated inall cases (Table II). Moreover, the total control effort used duringthe capture of the satellite is significantly reduced. In particu-lar, in the unconstrained controller, the control inputs saturatedwhenever the second and the third joints were not at their tar-get values. In the constrained controller the inputs only becamesaturated when the joint limits were reached. If the joint limitswere formulated into the control problem then the controllerwould be aware of the limit that it has reached, and would notcommand the joint to extend any further.
VII. CONCLUSION
This paper investigated the use of model-based predictivecontrol for the capture of a multi-(DOF) object that moves ina somewhat arbitrary manner, using a deployable manipulator.Predictive control was shown to be a feasible method for au-tonomously guiding a prototype manipulator toward a targetsatellite. Moreover, the feasibility of the constrained controllerwas demonstrated in both simulation and experimentation on aground based prototype of the MDMS.
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Richard A. McCourt received the Bachelor’s de-gree in engineering physics and the Master’s degreein mechanical engineering both from the Universityof British Columbia, Vancouver, BC, Canada, in 2002and 2004, respectively.
He is currently a Defence Scientist in the Centerfor Operational Research, Defence Research and De-velopment, Ottawa, ON, Canada.
Clarence W. de Silva (S’75–M’78–SM’85–F’98)received the Ph.D. degrees in dynamics systems andcontrol from the Massachusetts Institute of Technol-ogy, Cambridge and the University of Cambridge,Cambridge, U.K., in 1978 and1998, respectively.
Since 1988, he has been a Professor of mechani-cal engineering and holder of the NSERC-BC PackersChair in Industrial Automation at the University ofBritish Columbia, Vancouver, BC Canada. He wasthe Mobil Endowed Professor in the Department ofElectrical and Computer Engineering, National Uni-
versity of Singapore, Singapore. He is the author of 16 books, 165 journal papers,and 190 conference papers, including Sensors and Actuators: Control SystemInstrumentation (Taylor & Francis/CRC, 2007); Vibration—Fundamentals andPractice, 2nd ed. (Taylor & Francis/CRC, 2006); Mechatronics—An IntegratedApproach (Taylor & Francis/CRC, 2005); and Soft Computing and IntelligentSystems Design—Theory, Tools, and Applications (Addison-Wesley, 2004).
Dr. de Silva is a Lilly Fellow of Carnegie Mellon University, a Fellow ofNASA/American Society of Engineering Education (ASEE), Advanced Sys-tems Institute of British Columbia (ASI), Killam, Canadian Academy of En-gineering, a Senior Fulbright Fellow with the University of Cambridge, and aregistered Professional Engineer. He was on the Editorial Boards of 12 interna-tional journals including IEEE and ASME TRANSACTIONS an Editor-in-Chief ofthe International Journal, Control and Intelligent Systems, International Jour-nal of Knowledge-Based Intelligent Engineering Systems, the Regional Editorof IFAC International Journal—Engineering Applications of Artificial Intelli-gence, and a Senior Technical Editor of Measurements and Control. He was therecipient of the Henry M. Paynter Outstanding Investigator Award, the KillamResearch Prize, the Education Award of the ASME Dynamic Systems and Con-trol Division, the Outstanding Engineering Educator Award of IEEE Canada,the Outstanding Contribution Award of the IEEE SMC Society, and the Mer-itorious Achievement Award of the Association of Professional Engineers ofBritish Columbia.
