Internal Preload Control of Redundantly Actuated Parallel ...

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IEEE TRANSACTIONS ON ROBOTICS, VOL. 21, NO. 4, AUGUST 2005

Internal Preload Control of Redundantly Actuated Parallel Manipulators—Its Application to Backlash Avoiding Control A. Müller, Member, IEEE

Abstract—Redundant actuation of parallel manipulators can lead to internal forces without generating end-effector forces (preload). Preload can be controlled in order to prevent backlash during the manipulator motion. Such control is based on the inverse dynamics. The general solution of the inverse dynamics of redundantly actuated parallel manipulators is given. For the special case of simple overactuation an explicit solution is derived in terms of a single preload parameter. With this formulation a computational efficient open-loop preload control is developed and applied to the elimination of backlash. Its simplicity makes it applicable in real-time applications. Results are given for a planar 4RRR manipulator and a spatial heptapod. Index Terms—Backlash, inverse dynamics, model-based control, parallel manipulator, redundant actuation, tendon driven.

I. INTRODUCTION

C

OMMONLY used parallel manipulators (PMs) are nonredundantly full-actuated. That is, the PM is equipped with exactly as much actuated joints as its degree of freedom (DOF) is. There is an increasing interest in redundantly full-actuated (RFA) PMs. A number of redundantly actuated parallel mechanisms for use in robotic systems was proposed, such as robot hands [15], wrist [13] and shoulders [37]. In recent years redundant actuation has reached the field of robotic manipulators and machine tools. Examples for redundantly actuated PMs are the planar 3RRR PM in [11] with DOF 2, the 4RRR PM in [32] with DOF 3 and the spatial octapod in [32]. The first two are simply RFA, while the octapod has two redundant actuators. The term octapod refers to a hexapod with two additional struts. From a kinematic point of view redundant actuation eliminates singularities [25], [30] and so enlarges the usable workspace. Redundant actuation increases the payload and acceleration, can yield an optimal load distribution among the actuators or can reduce the power consumption [23]. Actuator redundancy can also improve the transmission properties. As such it increases the homogeneity of the force transmission and the manipulator stiffness (active stiffness) [36], [37]. A peculiar feature is the possibility of permanent control forces without generating end-effector (EE) forces. This peculiarity will be called internal preload. Preload can be employed for the control of elastic Manuscript received July 6, 2004; revised November 10, 2004. This paper was recommended for publication by Associate Editor J. P. Merlet and Editor F. Park upon evaluation of the reviewers comments. The author is with the Institute of Mechatronics at the Chemnitz University of Technology, 09126 Chemnitz, Germany (e-mail: [email protected]). Digital Object Identifier 10.1109/TRO.2004.842341

deflections [12] and for the elimination of backlash [32]. Backlash is due to joint clearing and DC motor hysteresis. The advantages of redundant actuation are, however, owed to demanding control and calibration methods [11], [33]. Backlash avoiding control in particular calls for a reliable preload control. General robust control schemes consist in a nonlinear feed forward control and a linear feedback regulation. This paper addresses the nonlinear feed forward term, i.e., the inverse dynamics, of RFA PMs in conjunction with backlash avoidance conditions. Optimal control is not the subject of this paper, rather the goal EE trajectory is assumed to be given from the path planning. A condition for backlash free motion is the absence of sign reversals of control the forces. Adding the requirement that all control forces be positive leads immediately to the definition of admissible postures of tendon driven manipulators. Due to this affinity the presented work is directly applicable to tendon driven platforms [9], [31], [34]. The paper is organized as follows. Section II gives a general formulation of the kinematics of holonomic constrained mechanical systems. This is used in Section III to derive the Lagrangian motion equations in terms of a set of independent generalized coordinates. Before considering the inverse dynamics a precise definition of actuator redundancy is given in Section IV. Section V addresses the inverse dynamics for RFA manipulators. An explicit solution for simple actuator redundancy is derived. In Section VI the developed formulations are incorporated in backlash avoiding preload control strategies. These strategies minimize the overall control forces such that backlash is prevented during the motion, that is a constrained optimization problem. The result is the time evolution of the preload for a given goal trajectory. The optimization problem is either solved at a finite number of sample points on the goal trajectory or the preload is approximated as a function of time. For a simply RFA PM the resulting optimization problem is one-dimensional. II. MANIPULATOR KINEMATICS Robotic manipulators are considered as multibody systems (MBS) composed of rigid bodies. A MBS subject to holonomic (i.e., geometric) constraints is a holonomic constrained mechanical system [17]. According to the existence of kinematic loops, robotic manipulators can be roughly classified as serial and parallel manipulators. While serial manipulators do not contain kinematic loops the presence of at least one loop is the characteristic of PMs. However, not every MBS with kinematic loops is a PM. Moreover several classes of PMs have to be distinguished.

1552-3098/$20.00 © 2005 IEEE

MÜLLER: INTERNAL PRELOAD CONTROL OF REDUNDANTLY ACTUATED PARALLEL MANIPULATORS

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According to the IFToMM terminology [7] a PM is a manipulator in which the end-effector (EE) is controlled by means of at least two kinematic chains going from the base to the EE. In the common use of the word, however, the term PM is used for fully parallel manipulators, according to IFToMM terminology. The latter are PMs with a degree of freedom1 (DOF) in which the EE is connected to the base by independent kinematic chains (limbs, struts, legs) each containing a single actuated joint. All limbs of a fully parallel manipulator are identical which facilitates their mass production. PMs are hence holonomic constrained MBS, where the constraints arise from the closure conditions of selected cut joints, i.e., kinematic loops are cut open and the corresponding closure conditions constitute a system of holonomic constraints.

(4)

A. MBS With Kinematic Tree Structure Each body of a MBS with kinematic tree structure is connected to the base by a unique kinematic chain. Without loss of generality adjacent bodies are assumed to be connected by either revolute, prismatic or screw joints. Let the PM contain rigid bodies. The configuration of body is expressed by (1) describes the relative rotation of a referTherein ence frame attached to body w.r.t. the inertial frame (world is the position vector of the reference frame frame). origin expressed in the IFR. The configuration (1) is given by the product of exponentials [19], [21]. The joint variables are generalized coordinates of the MBS. The configuration space of revolute and prismatic/screw joints is a MBS containing . The representing point is called the configuration of the MBS. The velocity of body is determined by the generalized velocities via (2) where the kinematic basic function of body body Jacobian)

The totality amounts to geometric constraints. Time differentiation yields the kinematic and dynamic constraints. The overall systems of geometric, kinematic, and dynamic constraints are

Without loss of generality the constraints are assumed to be, at least locally, independent, so that has full rank. The con, figuration space of the PM is which is an analytic variety and locally a smooth manifold, exis the set cept at singularities, where the rank of drops. of all admissible configurations of the PM. The tangent space is the vector space of all admissible velocities of the PM in the configuration . The DOF of , i.e., the local dimension of . According the PM is to the implicit function theorem applied to the geometric conof joint variables determines the configstraints a subset uration. There exists a smooth mapping , that assigns to each the PM configuration . The joint variables in are considered as independent generalized coordinates and as dependent, so that the remaining variables in the vector . Actually the map is a parametrization of the -dimensional manifold according to a local chart. Being a local parameterization means that is only bijective in a neigh, but not globally. E.g., the prismatic joint borhood of variables as independent coordinates of a Hexapod locally determine its configuration uniquely. However, it is well known that the forward kinematics has several different solutions for a particular posture of the prismatic joints. The independent velocities completely determine the generalized velocity of the PM by (5) in With the partition of kinematic constraints read (see e.g., [2])

the

(also called the Therewith follows that (3)

is given in terms of the instantaneous joint screws [3], [19]. is respectively the angular and linear velocity of body expressed in its reference frame.

(6) which is an orthogonal complement of , since accelerations follow with

. The

B. Holonomic Constrained Mechanical Systems Kinematic loops of the PM are cut open by selecting a cut joint for each fundamental loop. Each of the cut joints yields a certain number of geometric constraints, for example, a spherical joint contributes three translation constraints whereas a revolute joint gives three translation and two rotation constraints. 1Here the DOF of the MBS means the local DOF, which is to be distinguished

from the differential and global DOF, that is, the local dimension of the configuration space.

(7) Throughout the paper it is assumed that all matrices have constant rank in an open neighborhood of the considered point. C. End-Effector Constraints The task of a PM is to position an EE mounted on a certain body which is connected to the ground via a unique kinematic

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(9) can be projected onto the CMS configuration space using the orthogonal complement and the relations (6) and (7)

TABLE I SYMBOLS AND ABBREVIATIONS

(10) with

, where

These are the equations of Voronets [17]. Algorithmic aspects of their symbolic generations are considered in several papers, e.g., [27]. The system (10) together with the kinematic con, that comstraints (4) yields differential equations in pletely describe the MBS dynamics. Also the Lagrangian motion equations are, using the kinematic basic functions, given algebraically in closed form [3], [19]. IV. TERMINOLOGY FOR ACTUATED SYSTEMS

chain of the associated tree structured MBS. The EE velocity , where the matrix is reis determined by . ferred to as the EE-Jacobian. The local EE-DOF is If the EE-DOF is less then the DOF of the PM the manipulator is called kinematically redundant. The EE has to follow in workspace. The correa prescribed trajectory and responding goal EE velocity and acceleration is spectively. Using the IFR-independent distance measure on [21] the overall systems of geometric, kinematic and dynamic constraints are EE-constraints

A thorough classification of PMs in terms of their actuation is order. Actuation refers to the instantaneous relation between the input and the state derivative of a dynamical system in a given state. This is a pointwise property. The effect of the actuation is described by the controllability of the system. This is a local property, i.e., considering the effects over a small time [24]. The of a PM is with (10) determined by derivative of the state (11) and (7). The last term in (11) determines how the state derivative depends on the input . Let be the number of actuated joints. Then contains the corresponding generalized drive forces. Let be the submatrix of constituted by the columns corresponding to the actuated joints. Definition 1: The degree of actuation (DOA) is the dimension of the image space of

Closure constraints (8) It shall be emphasized that the kinematic basic functions (Jacobians) , the constraint Jacobian and their partial derivatives of any order can be given explicitly and algebraically in terms of screw products [3], [19]. III. MOTION EQUATIONS OF PARALLEL MANIPULATORS The dynamics of a force controlled holonomic constrained MBS with kinematic tree structure is governed by the Lagrangian motion equations (9) where is the generalized mass matrix, represents generalized Coriolis forces, contains generalized potential forces, represents all remaining forces (including EE forces) and are the generalized control forces. The Lagrange multipliers can be identified with the constraint reactions. The Lagrangian

If the PM is underactuated and if the PM is full-actuated. Definition 2: The degree of redundancy of the actuation is

The PM is called redundantly actuated if and nonredundantly actuated if . is -dimensional. In case of redundant acThe kernel of tuation nonzero control inputs exists which have no impact on the acceleration (preload). For the description of PMs the terms ‘overactuation’ and ‘redundant actuation’ are often used interchangeably. Redundant actuation refers to a control system of which the actual actuation could be achieved by a subset of the actual inputs. In control theory underactuation by definition refers to a control system which has less inputs then its DOF, while the number of inputs of a full-actuated system equals its DOF [29]. This is in accordance with Definition 1 because . A full-actuated system is completely actuated and an improvement is impossible. No other type of actuation beside


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be a desired preload vector, then (12) can be solved Let for such that

(13) is the pseudo--inverse and is a projector onto the -dimensional null of . If then and , space so that preload is impossible. In what follows the special but important case of simply reis considered, i.e., is a row vector. dundant actuation it holds that and the With the structure of pseudo-inverse of is where

Fig. 1. Different types of actuation for a 7-bar mechanism.

these two can exist. Hence the term ‘overactuation’ is meaningless and shall not be used. As an example, if the joints 1 and 2 of the 7-bar mechanism in Fig. 1 are actuated the PM is nonredundantly full-actuated. If joint 1, 2, and 3 are actuated the PM is redundantly full-actuated. Controlling joint 1 and 3 yields a redundantly underactuated PM. The remaining part of the paper deals with simply RFA PMs, , and so . i.e.,

For simple actuator redundancy the inverse always exists and the pseudo-inverse is found as (see Appendix) (14) Note that no matrix inversion is necessary. An orthogonal complement of is (15)

V. INVERSE DYNAMICS OF REDUNDANTLY FULL-ACTUATED PMS The inverse dynamics problem consists in finding the generalized control forces that produce a desired trajectory of the find such that (9) holds. PM. That is, given . The DOA of a full-actuated PM equals its DOF The number of active drives of a RFA PM exceeds its DOF by . Without loss of generality the vector consisting . of joint variables of the actuated joints is The vector of the remaining joint variables of passive joints . Accordingly the is denoted with , so that , where generalized control vector has the form comprises the generalized forces associated to the actuated joints. A subset of active joint variables constitutes a vector of independent coordinates, such that . This gives rise to a partition of the orthogonal complement

. Hence is a projector onto and the i.e., columns of are basis vectors for this null space. If then is a vector so that any null space component can be expressed , with a parameter . Substitution of (14) and (15) in as (13) yields (16) abbreviating . Thus the preload depends smoothly on a single preload paramand the acceleration of the PM the eter . Given a state corresponding controls are uniquely determined by via (16). Remark on preload free force balanced control: If backlash is not an issue redundant actuation allows to minimize the average control forces and thus the use of smaller drive is given motors. The inverse dynamics solution minimizing by (13), or particularly by (16), with . No preload is generated in this case. VI. BACKLASH AVOIDING INVERSE DYNAMICS

where

contains the first and the remaining rows of . If then , and vanishes. Projecting the motion equations (9) to the configuration space yields using (12)

The derived relations for the inverse dynamics of simply RFA PMs are incorporated in a backlash avoiding feed forward control scheme by which the PM is controlled along a prescribed . The trajectory is obtained by solving the inverse trajectory kinematics for the goal EE trajectory. The main idea is to include the generation of internal preload in the control scheme of the PM. Here the subject is not optimal control in the sense

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that optimal trajectories and the corresponding controls are determined, although this is a straightforward extension of the presented work. Rather the EE trajectory is assumed to be given as result of the preceding path planning.

termined by the goal EE-trajectory via the inverse kinematics the objective attains a minimum iff is minimized for all . Therefore the backlash avoiding optimal control problem is

A. Problem Formulation Given a EE-trajectory , the corresponding trajectory in satisfying the EE- and configuration space is a solution the closure constraints (8), i.e., the inverse kinematics solution. are determined Controls that force the PM to move along by the inverse dynamics. The freedom in generating preload is used to prevent backlash. A condition for backlash free control is that the magnitude of each particular control force remains above a certain level and that its sign remains constant [33] during the considthe reered task with a duration . Denote with quired sign of , then the condition (17) must be satisfied with . There are possible combinations of the signs. If an admissible combinations is a priori whether there is a unknown it must be checked for all for which (17) together with (13) or (16) holds. It is worth to note the relation of RFA PMs with actuated prismatic joints and tendon driven platforms. The latter are controlled by pulling its tendons, and since tendons cannot exert pushing forces a permanent pretension is necessary (as in case of tensegrity structures). Identifying tendons and active prismatic joints all subsequent considerations are applicable. In this context a tendon driven simply RFA PM corresponds to a “Completely Restrained Positioning Mechanism”, while a tendon driven RFA PM with higher degree of redundancy is called “Redundantly Restrained Positioning Mechanism” [34]. Assume that positive controls correspond to pulling forces. , the condition (17) demands a sufficient Then, with tendon pretension. Consider for example a Gough-Steward platform with one additional strut, each strut consisting of a universal joint connecting it to the base, an actuated prismatic joint and a spherical joint as connection to the platform. If the control force in a prismatic joint always remains positive the corresponding strut can be replaced by a pulley mechanism. Eventually a construction can be found where all seven struts are replaced by cable drives (with reduced workspace, however) [9]. Due to redundant actuation the solution is not unique, and the freedom in choosing the control forces is used to achieve an op. The cost function may timum w.r.t. the objective serve for arbitrary goals. For instance the cost function expresses the drive power, whereas represents the overall control forces. The positive definite weighting matrix weighs different drive characteristics. Beside these dynamic evaluations also kinematic considerations can constitute reliable metrics. As such kinematic dexterity measures are well established criterions for the evaluation of robotic manipulators [5], [18], [21], [26]. The peculiarity of nonreversing control forces demands, however, a redefinition of those dexterity measures [14]. Since the system dynamics is completely de-

with being the maximum load magnitude of drive . This subject to linear is a quadratic optimization problem in equality and inequality constraints, where the linear equality constraints are time dependent. Remark on inverse kinematics: Generally there is no closed solution of the inverse kinematics. At discrete time instants the configuration solving the geois found by numermetric EE-constraints ical continuation. The corresponding velocity and acceleration are then found from

B. Optimal Control Forces for Simply Redundantly Full-Actuated PM According to (13) the general control problem (BAC) in could be transformed in a control terms of the controls problem in terms of the preload . But the number of controls remains . For simply redundant actuation, however, the general problem (BAC) can be reduced to the one-dimensional optimization problem in the function (18) with in (16). Thus the task is to find the time evolution of the preload parameter minimizing the objective while satisfying the sign convention and force limitations (17). If the optimal control aims to minimize control forces and if the PM is , then . equipped with identical drives, i.e., In this case and , so that

Since the first term only depends on the goal trajectory the objective attains a minimum w.r.t. to iff the second term attains a minimum. Therewith arises the advantageous form (19) with in (16). In summary a function has to be determined that solves the control problem (18), or (19) if control


forces are to be minimized. There are basically two numerical approaches: either the function is approximated or its values at discrete time instants are determined. The first approach yields an explicit formulation of the preload parameter as a function of time, while the second only yields its numerical values at specific time instants. The latter is not necessarily problematic, moreover it is in accordance with technical applications, where the system is only evaluated with a finite sampling rate. 1) Pointwise Determination of Optimal Control Forces: Let be discrete time instants, with . At any time instant the constraints in (BAC) determine an at most -dimensional subspace of admissible con). For simply RFA PMs this trols (actually a subspace of subspace is at most one-dimensional, if not empty at all. The is a valid parameterization of preload parameter this solution space. At the formulation (18) and (19) constitute respectively a convex one-dimensional quadratic optimization problem in the unknown . A look at the constraints reveals that they constitute simple bounds on due to (16) which is linear in . The instantaneous minimization problems could be solved with adequate solvers for constrained NLP problems. However, a solution can be found with a computational inexfrom the preceding time step as pensive line search using initial value. But the choice of the initial preload parameter is crucial, and at the one dimensional search must consider can the complete range of . The preload increment be limited in order to restrict control jerks. Generally will be a local minimum even if the PM starts with a globally optimal at . were assumed to be known. Not all So far suitable signs possible combinations will allow preload. In order of the to find the sign vector that enables backlash avoiding control possible combinations with minimal control forces all of the of the signs must be checked. Thus the computational effort . That is, the complete control task increases by the factor must be considered for sign combinations in order to ensure that the preload condition can be satisfied during the task execution using a specific sign vector. Due to its simplicity the proposed method for pointwise computation of the control forces is tailored for real time applications. However, the sign convention will generally have to be determined upon global considerations. In time critical applications in order to lighten the computational effort, the first admissible choice that is foundcan be used. 2) Global Approximation of Optimal Control Forces: The pointwise determination of the preload that minimizes the control forces and ensures the preload conditions (17) can lead to a erratic behavior of the control forces. This effect can be circumvented by using a smooth approximation function for . The preload function is approximated by an -segment spline

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only coefficients of are w.r.t. parameter changes independent. The sensitivity of as independent suggests to choose parameters. Define , and . With the original objective the integral form of (18) is (21) after replacing in (16). Analogously with (19) if . This is a nonlinear optimization problem in the spline variables subject to two equality constraints. The constraints are only continuous but not continuously differentiable . The NLP problem (21) is solved with standard condue to strained NLP methods [4], [28]. If only continuity of is required the cubic polynomials are replaced by linear functions. Also alternative trial functions can be used for such as B-splines or trigonometric polynomials. The solution of (21) is rather time consuming and not applicable in real time application. It should be emphasized that although the preload parameter is approximated the inverse dynamics solution remains exact. The only difference to the pointwise minimization is that the control forces are not completely minimized. with

C. Feed Forward Control Scheme for FRA PMs The above inverse dynamics solution delivers the feed forward term in a nonlinear controller. Applying the feed forward control will generally not yield the desired motion due to model uncertainties and effects related to measurement errors or time discretization. A variety of control schemes can cope with these effects. Regardless of the control approach a methodology for incorporating preload in general nonlinear control schemes for RFA PMs is proposed. suggests to split the control vector The form . Therein are controls correaccordingly: and is the vector sponding to the independent variables of the redundant control forces. The structure of allows to write (10) as

Having determined the strategy is to consider the excess conas given generalized forces. A nonlinear controller trols whatsoever (continuous,discrete) will use the inverse dynamics solution as nonlinear feed forward term. The linear feedback term will control but not interfere with the redundant actua. Therewith mutually conflicting actuation is prevented, tion which will always occur due to model uncertainties, e.g., the widespread computed-torque PD control law [21] becomes

(20) For a cubic spline interval ditions

are cubic polynomials defined on the . Due to the attachment conand

where respectively and is the desired and measured trajecis the error vector and and are gain tory, . matrices. The PM is controlled by

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Fig. 2.


Planar 4RRR manipulator with a goal EE path.

VII. EXAMPLES The presented approach was implemented in a MBS dynamics simulation tool. A variety of planar and spatial manipulators were investigated aiming to minimize the overall control forces. The first example is a 4RRR PM [32]. As its name suggests, the manipulator has 4 identical legs each consisting of a chain of three revolute joints connecting the base to the platform (Fig. 2). Since all joint axes are parallel the platform can only perform planar motions. The 4RRR PM originates from the 3RRR PM that has attracted much interest due its simple construction, but also due to the hazardous distribution of singularities in its workspace [6]. Most of the singularities of the 3RRR are removed by redundant actuation [32]. Thus the effort for the additional fourth chain is justified by the enlarged singularity free workspace and the ability to allow for components with joint clearing when controlling internal preload. Since closed-loop control is not the subject of this paper the model is assumed to be perfect and the manipulator is controlled by the inverse dynamics solution. The 4RRR PM has three kinematic loops. The loops are opened by removing three revolute joints connecting the respective limb to the platform, joint variables as indicated in Fig. 2. The remaining of the resulting kinematic tree structure constitute the generalized coordinates of the constrained MBS. Each of the three cut joints delivers two constraints constituting a system of 6 independent closure constraints. The DOF of . The joints at the base are actuated. the PM is The vector of active joint variables is of are considered as independent. The which configuration of the PM is uniquely determined by . That is with simple actuator redunthe system is full-actuated dancy . Fig. 2 depicts the position of the EE-tip for a desired motion. In this example the EE has to remain in its initial orientation. The EE-trajectory is determined according to velocity, acceleration and jerk limits. The corresponding time evolution of the joint variables, velocities and accelerations are found from the inverse kinematics (Fig. 3). Because of the four . An actuator preload identical drives the weighting is Nm is required. Upper bounds are not considered. of

Fig. 3. Joint coordinates and velocities of the 4RRR-PM when tracing the EE path in Fig. 2.

Fig. 4. Preload parameter and control torques for the EE motion in Fig. 2 with sign vector s = (1; 1; 1; 1)—pointwise solution.

0

0

The task duration of 6 s is discretized in 5 ms steps and (19) is solved with a local one-dimensional search. This is possible in real time without paying attention to an efficient computer possible choices implementation. The evaluation of all of the sign vector reveals that the required preload is only possible with and . Interestingly for all considered motions of the 4RRR only with or preload can be achieved. Fig. 4 shows the evolution of the preload parameter and the corresponding drive torques


Fig. 5. Preload parameter and control torques for the EE motion in Fig. 2 with sign vector s = (1; 1; 1; 1)—spline approximation.

0

0

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Fig. 7. Preload parameter and control torques for the EE motion in Fig. 2 with sign vector s = ( 1; 1; 1; 1)—spline approximation.

0

0

Fig. 8. 4RRR PM moving with EE orthogonally aligned to an ellipsoidal path.

Fig. 6. Preload parameter and control torques for the EE motion in Fig. 2 with sign vector s = ( 1; 1; 1; 1)—pointwise solution.

0

0

for . Obviously the control torques exactly trace the lower . This is due to the pointwise bound according to . The global formulation minimization of the objective (21) yields an approximate but smooth behavior. Fig. 5 shows segments. The problem results for a cubic spline with (21) is solved for the 23 independent coefficient with a standard method for constrained optimization [4], [28]. The applied algorithm converges rapidly but the necessary computation time makes a real time solution impossible. The admissible sign vector yields smaller control torques as shown in Figs. 6 and 7. The second task consists in following the surface of a cylindrical object with the EE orthogonally aligned to the

surface (Fig. 8). For this task the required preload of nm can only be achieved with . Results are shown in Fig. 9. The heptapod (a hexapod with one additional strut) in Fig. 10 serves as second example. It is closely related to the Falcon [9], a PM equipped with seven cable drives. The seven prismatic joints are actuated. For this spatial RFA PM the task was to draw a circle on a spherical surface with orthogonally aligned EE and without rotation about the longitudinal axis. The ask must be accomplishes in 10 s. There are possibilities, but the required preload of 100 N can only be achieved with , . Fig. 11 shows the pointwise or solution for the controls according to . Using only changes yields the optimum (Fig. 12). In the latter the signs while case , i.e., for this task the minimum norm solution that produces the required preload is the particular pseudo-inverse is obtained solution in (16). A smooth time evolutions of using the spline approximation (20).

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Fig. 9. Control torques for the EE motion in Fig. 8 with sign vector ( 1; 1; 1; 1)—pointwise solution.

0

0

s

=

Fig. 11.

s

Fig. 10.

Control forces c and preload parameter for the task in Fig. 10, with c = 100 N.

= (1; 1; 1; 1; 1; 1; 1) and a preload

Heptapod in its initial posture.

VIII. SUMMARY Redundant actuation permits internal forces (preload) without exerting EE-forces. The inverse dynamics of redundantly actuated parallel manipulators is addressed. The general solution gives rise to a computational very efficient formulation for simply redundantly actuated manipulators. In this formulation the preload depends on a single parameter . Based on this formulation a preload control problem is constructed. Its solution is the time evolution of the preload parameter , such that the required level of preload is ensured, and the overall actuator forces are minimized. With suitably chosen preload backlash is prevented. Two numerical approaches are proposed. The first approach solves a one-dimensional constrained minimization problem at a finite number of points on the trajectory. This is applicable in real time since it only requires a one dimensional continuation of the preload parameter. The second approach approximates the preload parameter as a function of time by a suitable trial functions, e.g., splines, resulting in a constrained NLP problem. This formulation can be extended to an optimal and where only the initial and final control problem in and but not the trajectory itself configurations is specified. This problem can be solved with direct methods of optimal control [1], [16], [20], [22], [35], i.e., using trial

Fig. 12. Control forces for the task depicted in Fig. 10, with 1; 1; 1; 1; 1; 1; 1) and c = 100 N. In this case 0.

(

0 0 0 0

s

=

functions for and for . Crucial for backlash avoidance is an appropriate choice of the required signs of the control forces. This remains a challenge since this choice is by its very nature based on the comparison of all admissible sign combinations. So further work necessarily has to address this particular issue. It might be suspected that the admissible signs are related to the particular kinematics and geometry of a PM. APPENDIX I PSEUDO-INVERSE OF The special case of simply redundant actuation , i.e., , is considered here. Denote with the eigenvalues of and respectively. always has rank 1, so that The matrix and . Because the only nonzero eigenvalue of is . are Thus the eigenvalues of


and positive definite with determinant With

This holds for

. Hence

is . the von-Neuman series yields

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A. Müller (M’97) received the Diploma degree in mathematics from the University of Applied Sciences, Mittweida, Germany, in 1997, the Diploma degree in mechanical engineering, and the Ph.D. degree in mechanics (with honors) from the University of Technology, Chemnitz, Germany in 2001and 2004, respectively. Since 1998, he has been with the Institute of Mechatronics, University of Technology. His research interests include dynamics and control of mechatronic systems, kinematics of mechanisms, singularities, differential geometric methods, advanced simulation methods, MEMS, and chaotic communications