Internal Preload Control of redundantly actuated Parallel Manipulators – Backlash avoiding Control A. M¨uller, MIEEE, Email:
[email protected] Institute of Mechatronics at the Chemnitz University of Technology Reichenhainer Straße 88, 09126 Chemnitz, Germany
Abstract— Redundant actuation of parallel manipulators admits 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 proposed for eliminating backlash. Its simplicity makes it applicable in real-time applications. The approach is demonstrated on a planar 4RRR manipulator. Index Terms— Redundant actuation, inverse dynamics, parallel manipulator, backlash, model based control
I. I NTRODUCTION In recent years redundant actuation has reached the field of robotic manipulators and machine tools. A number of redundantly actuated parallel mechanisms for use in robotic systems was proposed, such as robot hands [9], wrist [8] and shoulders [23]. Examples for redundantly actuated parallel manipulators (PM) are the planar 3RRR PM in [6] with degree of freedom (DOF) 2, the 4RRR PM in [20] with DOF 3 and the spatial octapod in [20]. The first two are simply redundantly full-actuated (RFA), while the octapod has two redundant actuators. The term octapod refers to a hexapod with 2 additional struts. From a kinematic point of view redundant actuation eliminates singularities [16],[19] 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 [14]. Actuator redundancy can also improve the transmission properties. As such it increases the homogeneity of the force transmission and the manipulator stiffness (active stiffness) [22],[23]. 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 deflections [7] and for the elimination of backlash [20]. Backlash is critical in the presence of joint clearing and DC motor hysteresis. The advantages of redundant actuation are, however, owed to demanding control and calibration methods [6],[21]. 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.
II. M ANIPULATOR KINEMATICS In the common use of the word aPM is a controlled manipulator in which the EE is connected to the base by several kinematic chains (limbs, struts, legs) containing actuated joint. The limbs may be identical which facilitates their mass production. PM are hence holonomic constrained mechanisms. 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. A. Holonomic constrained mechanical systems Without loss of generality adjacent bodies are assumed to be connected by DOF 1 joints, i.e. either by revolute, prismatic or screw joints. Kinematic loops of the PM are cut open by selecting a cut joint for each fundamental loop of the topology graph [1],[11]. The joint variables q a of the remaining joints of the kinematic tree structure constitute the generalized coordinates of the PM. The configuration space of the PM with nR revolute and nP prismatic/screw joints is a subspace of Vn := TnR × RnP , n = nR + nP . The representing point q ≡ (q a ) ∈ Vn is called the configuration of the PM. Each of the cut joints gives rise to a certain number of geometric constraints. E.g. a spherical joint contributes 3 translational constraints and a revolute joint gives 3 translational and 2 rotational constraints. The totality amounts to r geometric constraints. Time differentiation yields the kinematic and dynamic constraints. The overall systems of geometric, kinematic and dynamic constraints are 0 0 0
= h (q) , h ∈ Rr ˙ f ∈ Rr,n = f (q) q, ¨ + f˙ (q) q. ˙ = f (q) q
(1)
Without loss of generality the r constraints are assumed to be, at least locally, independent, so that f has full rank. Thereupon the configuration space of the PM is V = {q ∈ Vn |h (q) = 0}, which is an analytic variety and locally a smooth manifold except at singularities, where the rank of f drops. V is the set of all admissible configurations of the PM. The local DOF of the PM is δ = n − r, i.e. the local dimension of V . According to the implicit function theorem applied to the geometric constraints a subset q2 comprised of δ joint variables determines the configuration. There exists a smooth mapping ψ, that assigns to each q2 the PM configuration q a = ψ a (q2 ). The joint variables in
q2 are considered as independent generalized coordinates and the remaining r variables in the vector q1 as dependent, so that q = (q1 , q2 ). Actually the map ψ −1 is a parameterization of the δ-dimensional manifold V according to a local chart. Being a local parameterization means that ψ is only bijective in a neighborhood of q ∈ V , but not globally. E.g. the prismatic joint 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 ∂ a ˙q = Fq˙ 2 , where F := ψ . (2) ∂q b With the partition of f = (f1 , f2 ) in f1 ∈ Rr,r , f2 ∈ Rr,δ the kinematic constraints reads f1 q˙ 1 + f2 q˙ 2 = 0. Therewith follows that F=
−f1−1 f2 Iδ
!
∈ Rn,δ ,
EQUATIONS OF PARALLEL MANIPULATORS
The dynamics of a force controlled holonomic constrained MBS with kinematic tree structure is governed by the Lagrangian motion equations (LE) ¨ + C (q, q) ˙ q˙ + N (q) + Q (q, q, ˙ t) + f T (q) λ = u, G (q) q (5) where G is the generalized mass matrix, Cq˙ represents generalized Coriolis forces, N contains generalized potential forces, Q represents all remaining forces and u are the generalized control forces. The Lagrange multipliers λ can be identified with the constraint reactions. The LE (5) can be projected onto the CMS configuration space V using the orthogonal complement F and the relations (3) and (4): T
¨ 2 + C (q, q) ˙ q˙ 2 + N (q) + Q (q, q, ˙ t) = F u, (6) G (q) q with q˙ = Fq˙ 2 , where ˙ N:=FT N, Q:=FT Q. G:=FT GF, C:=FT (CF + GF), These are the equations of Woronetz [10]. The system (6) together with the kinematic constraints in (1) yield n differential equations in q ∈ Vn , that completely describe the MBS dynamics.
FOR ACTUATED SYSTEMS
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 ˙ of a PM small time [15]. The derivative of the state (q, q) is with (6) determined by −1 −1 ¨ 2 = −G q Cq˙ 2 + N + Q + G FT u (7)
and (4). This is a nonlinear control system, where the first term on the right hand side is the drift vector field. The last term in (7) determines how the state derivative depends on the input u, in the actual configuration. Let m of the joints be actuated. Then u contains the corresponding m generalized drive forces. Let M ∈ Rδ,m be the submatrix −1 of G FT constituted by the m columns corresponding to the actuated joints. Definition 1: The degree of actuation (DOA) is the dimension of the image space of M α := rank (M) .
(3)
which is an orthogonal complement of f , since fF ≡ 0. The accelerations follow with ! f1−1 (f˙1 f1−1 f2 − f˙2 ) ˙ ˙ ¨ = F¨ q q2 + Fq˙ 2 , F = ∈ Rn,δ . 0 (4) Throughout the paper all matrices are assumed to have constant rank in open neighborhood of the considered point. III. M OTION
IV. T ERMINOLOGY
If α < δ the PM is underactuated and if α = δ the PM is full-actuated. Definition 2: The degree of redundancy of the actuation is ρ := m − α. The PM is called redundantly actuated if ρ > 0 and nonredundantly actuated if ρ = 0. The kernel of M is ρ-dimensional. In case of redundant actuation nonzero control inputs exists which have no impact on the acceleration (preload).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 [18]. This is in accordance with definition 1 because α ≤ m. For the description of PMs the terms ’overactuation’ and ’redundant actuation’ are often used interchangeably. A full-actuated system is completely actuated and an improvement is impossible. So the term ’overactuation’ is meaningless. In what follows the focus is on simply redundantly full-actuated (RFA) PMs, i.e. m = δ + 1, α = δ and so ρ = 1. V. I NVERSE
DYNAMICS OF REDUNDANTLY
FULL - ACTUATED
PM S
The inverse dynamics problem consists in finding the generalized control forces u that produce a desired trajectory of the PM. That is, given q (t) find u (t) such that (5) holds. The DOA of a full-actuated PM equals its DOF (δ = α). The number m = δ + ρ of active drives of a FRA PM exceeds its DOF by ρ. Without loss of generality the vector with joint variables of the actuated joints is qa = q 1 , . . . , q m . The vector of remaining joint variables
of passive joints is denoted with qp , so that q = (qp , qa ). Accordingly the generalized control vector has the form T T u = (0, c) , where c ≡ (c1 , . . . , cm ) comprises the generalized forces associated to the actuated joints. A subset of δ active joint variables constitutes a vector of independent coordinates, such that qa ≡ (. . . , q2 ). This gives rise to a partition of the orthogonal complement P −f1−1 f2 F = ≡ , A Iδ A1 P ∈ Rn−m,δ , A = ∈ Rm,δ , A1 ∈ Rρ,δ , Iδ
where P contains the first n − m and A1 the remaining rows of −f1−1 f2 . If ρ = 0 then P ≡ −f1−1 f2 , A ≡ Iδ and A1 vanishes. Projecting the LE (5) on the configuration space using FT yields
q¨ of the PM the corresponding controls c are uniquely determined by σ via (12). Remark on preload free force balanced control: If backlash is not an issue, redundant actuation allows to minimize the average driving force and thus the use of smaller drive motors. The inverse dynamics solution minimizing cT c is given by (9), or specifically by (12), with σ ≡ 0. No preload is generated in this case. VI. BACKLASH
AVOIDING INVERSE DYNAMICS
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 trajectory q (t). 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 that optimal trajectories and the corresponding controls are determined, even though this is a straightforward extension of the presented work. Rather the EE trajectory is assumed to be given from a preceding path planning.
¨ + C (q, q) ˙ q+N ˙ ˙ = AT c. FT (G (q) q (q) + Q (q, q)) (8) Let c0 ∈ Rm be a desired preload vector, then (8) can be solved for c such that ||c − c0 || → min : + ¨ + C (q, q) ˙ q˙ + N (q) + Q (q, q)) ˙ c = AT FT (G (q) q A. Problem formulation + T T 0 +(Im − A A )c . (9) Controls that force the PM to move along the goal q (t) + −1 are known from the inverse dynamics. The freedom in AT := A AT A is the pseudoinverse and generating preload is used to prevent backlash. A condition T + T (Im − A A ) is a projector onto the ρ -dimensional for backlash free control is that the magnitude of each T T null space of A . If m = δ then A ≡ Iδ and N A particular control force ca remains above a certain level T N A = ∅, so that no preload is possible. cmin and that its sign remains constant [21] during the a In what follows the special but important case of simply considered task with a duration T . Denote with sa ∈ redundant actuation (ρ = 1) is considered, i.e. A1 is a row {−1, 1} the required sign of ca then the condition vector. With the structure of A it holds that rank(A) ≡ δ T and the pseudoinverse of A is sa ca (t) ≥ cmin (13) a , t ∈ [0, T ] −1 + A1 AT = AT1 A1 + Iδ . must be satisfied. There are 2m possible combinations of Iδ the m signs. If an admissible choice is not known it must be For simple actuator redundancy the inverse always exists checked for all 2m whether there is a σ (t) for which (13) and the pseudoinverse is found as (see appendix) in conjunction with (9) or (12) holds. Due to redundant actuation the solution is not unique and the freedom in + A1 T 1 . (10) AT = Iδ − 1+kA 2 A1 A1 choosing the control forces is used to achieve an optimum 1k Iδ in a certain sense. The requirement to the overall Z minimize T Note that no matrix inversion is necessary. An orthogonal control forces is expressed by J = cT Wc dt → min. complement of AT is 0 The positive definite weighting matrix W ∈ Rm,m weighs Iρ m,ρ B= R , (11) different drive characteristics. Since the system dynamics −AT1 is completely determined by the goal EE-trajectory via the i.e. AT B = 0. Hence B is a projector onto N AT and the inverse kinematics the objective J attains a minimum iff columns of B are basis vectors for this null space. If ρ = 1 the integrand is minimized for all t ∈ [0, T ]. Therefore the then B is a vector so that any null space component can backlash avoiding optimal control problem is be expressed as σB, with a parameter σ ∈ R. Substitution T c Wc → min, t ∈ [0, T ] of (10) and (11) in (9) yields AT c = ϕ T A1 A1 A1 (BAC) min c= Iδ − 1+kA k2 ϕ + σB, (12) sa ca ≥ ca 1 Iδ sa ca ≤ cmax a ¨ 2 + C (q, q) ˙ q˙ 2 + N (q) + abbreviating ϕ := G (q) q ˙ Q (q, q). This is a quadratic optimization problem in c (t) subject to Thus the preload depends smoothly on a single preload linear equality and inequality constraints, where the linear ˙ and the acceleration parameter σ. Given a state (q, q) equality constraints are time dependent.
B. Optimal control forces for simply redundantly full-actuated PM According to (9) the general control problem (BAC) in terms of the controls c could be transformed in a control problem in terms of the preload c0 . But the number of controls remains m. For simply redundant actuation, however, the general problem (BAC) can be reduced to the one-dimensional optimization problem in the function σ T c Wc → min, t ∈ [0, T ] min (14) sa ca ≥ ca max sa ca ≤ ca
with c (σ) in (12). Thus the task is to find the time evolution of the preload parameter σ minimizing the objective cT Wc while satisfying the sign convention and force limitations (13). If the PM is equipped with identical drives then W BT B = ≡ TIm+. Inthis case AT 2 T + T 1 A1 1+kA1 k , A A = Iδ − 1+kA k2 and 1 + BT AT = 0, so that AT 2 1 A1 cT c = ϕT Iδ − 1+kA ϕ + 1 + kA1 k σ 2 . 2 k 1
Since the first term only depends on the goal trajectory q (t) the objective attains a minimum w.r.t. to σ (t) iff the second term attains a minimum. Therewith arises the advantageous form 2 2 1 + kA1 k σ → min, t ∈ [0, T ] min , (15) s c ≥ c a a a sa ca ≤ cmax a with c (σ) in (12). In summary a function σ has to be determined that minimizes the respective objective in (14) or (15). Let ti > ti−1 , i = 0, . . . , N be discrete time instants, with t0 = 0, T = tN . At time instant ti the constraints in (BAC) determine a ρ-dimensional subspace of admissible controls. For simply RFA PMs this subspace is at most one-dimensional, if not empty at all. The preload parameter σi := σ (ti ) is a valid parameterization of this solution space. At ti the formulation (14) and (15) constitute respectively a convex one-dimensional quadratic optimization problem in the unknown σi . A look at the structure of the constraints reveals that they constitute simple bounds on σi due to (12) which is linear in σ. The instantaneous minimization problems could be solved with adequate solvers for constrained NLP problems. However, a solution can be found by a computational inexpensive line search using σi−1 of the preceding time step as initial value. But the choice of the initial preload parameter σ0 is crucial, and at t0 the one dimensional search must consider the complete range of σ. The preload increment |σi − σi−1 | can be limited in order to avoid undesirable jerks in σ. Generally σi will be a local minimum even if the PM starts with a globally optimal σ0 at t0 . So far proper signs sa were assumed to be known. Not all of the 2m possible combinations will allow preload. In
order to find the sign vector s that enables backlash avoiding control with minimal control forces all 2m possible combinations of the m signs must be examined. Thus the computational effort increases by the factor 2m . That is, the complete task must be considered for 2m 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, to lighten the computational effort, the first admissible choice that is found can be used. C. Control scheme for redundantly full-actuated PM The above inverse dynamics solution delivers the feed forward term in a nonlinear controller. Applying the feed forward controls will generally not yield the desired motion due model uncertainties and effects related to measurement errors or time discretization. A variety of control schemes can cope with these effects [13],[21]. Regardless of the control approach a methodology for incorporating preload in general nonlinear control schemes for RFA PMs is proposed. The control vector is split according to c = (cex , c2 ). Therein c2 are controls corresponding to the independent variables q2 and cex is the vector of the ρ redundant control forces. The structure of F admits to write (6) as T cex c2 = G¨ q2 + Cq˙ 2 + N + Q + f1−1 f2 . 0
Having determined c, the strategy is to consider the excess controls cex as given generalized forces. A nonlinear controller whatsoever (continuous, discrete) will use the inverse dynamics solution c as nonlinear feed forward term. The linear feedback term will control c2 but not interfere with the redundant actuation cex . Therewith mutually conflicting actuation is prevented, which will always occur due to model uncertainties. E.g. the widespread computedtorque PD control [13] becomes T cex e c2 = G¨ qd2 +Cq˙ 2 +N+Q−G (Kv e˙ 2 + Kp e2 )+ f1−1 f2 , 0
where qd and q is the desired and measured trajectory respectively, e := qd − q is the error vector function and Kv and Kp are gain matrices. The PM is controlled by e c = (e c2 , cex ). VII. E XAMPLES
Results are reported for the inverse dynamics of a 4RRR PM [20]. The corresponding control is not considered here. The manipulator has 4 identical legs each consisting of a chain of three revolute joints connecting the base to the platform, as shown in figure 1. 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 to its simple construction but also due to the hazardous distribution of singularities in its workspace
2.5
q4 q5 [Nm]
2
cut joint 2 EE-path
q7
q1
1.5
q3 1
cut joint 3
q6
q2
cut joint 1
0.5
q9 3
q8 2
Nm
c2 c4
1
Fig. 1.
Planar 4RRR manipulator with a goal EE path.
0
-1
0.6
c1 c3
1
q q2 q3 q4 q5 q6 q7 q8 q9
0.4
rad
0.2
0
-0.2
-2 0
1
2
t [s]
3
5
4
6
Fig. 3. Preload parameter and control tourques for the EE motion in figure 1 with sign vector s′ = (1, −1, 1, −1). -0.2
-0.4
-0.3
q_ 1 q_ 2 q_ 3 q_ 4 q_ 5 q_ 6 q_ 7 q_ 8 q_ 9
rad/s
0.5
0
-0.5
-1
[Nm]
1
-0.4 -0.5 -0.6
-0.7 -0.8 -0.9
0
1
2
3
t [s]
4
5
6
1
Fig. 2. Joint coordinates and velocities of the 4RRR-PM when tracing the EE path in figure 1.
c3 c1
Nm
0.5
0
[3]. Most of the singularities of the 3RRR are removed by redundant actuation. 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. The 4RRR PM has three kinematic loops. The loops are opened by removing three revolute joints connecting the respective limb to the platform, as indicated in figure 1. The remaining n = 9 joint variables q a , a = 1, . . . , 9 constitute the generalized coordinates of the kinematic tree structure. Each of the three cut joints delivers two constraints constituting a system of 6 independent closure constraints. Thus the DOF of the PM is δ = 3. The m = 4 joints at the base are actuated. The vector of active joint variables is qa = q 1 , q 4 , q 6 , q 8 of which q2 = q 1 , q 4 , q 6 are considered as independent generalized coordinates. The configuration of the PM is uniquely determined by qa .
-0.5
c2 -1
c4 0
1
2
t [s]
3
4
5
6
Fig. 4. Preload parameter and control tourques for the EE motion in figure 1 with sign vector s′′ = (−1, 1, −1, 1).
That is the system is full-actuated (α = δ) with simple actuator redundancy ρ = m − α = 1. Figure 1 depicts the position of the EE-tip for the desired motion with a fixed EE-rotation. The EE-trajectory is planned according to the maximal velocity, acceleration and jerk. Because of the four identical drives the weighting is W = I4 . An
actuator preload magnitude of cmin = 0.2 Nm is required. a Upper bounds are not considered. The task duration of 6 s is discretized in 5 ms steps and (15) is solved with a local onedimensional search. This is possible in real time without paying attention to an efficient computer implementation. The evaluation of all 24 possible choices of the sign vector s reveals that the required preload is only possible with s′ = (−1, 1, −1, 1) and s′′ = (1, −1, 1, −1). Interestingly for all considered motions of the 4RRR no other then s′ and s′′ may achieve preload. Figure 3 shows the evolution of the preload parameter σ and the corresponding drive torques for s′ , which yields lower overal control tourqes then s′′ . The control torques exactly trace the lower bound of cT c according to ca ≥ cmin and the time discretization. a VIII. S UMMARY 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 σ. Upon this formulation a preload control problem is formulated. Its solution is the time evolution of the preload parameter σ such that the required level of preload is ensured and the overall actuator forces are minimized. Backlash is prevented with suitably chosen preload. At finite time instants (according to a specific sampling rate) the problem is considered as one dimensional constrained minimization problem. Thus the problem may be solved in real time since it only requires a one dimensional continuation of the preload parameter. Crucial for backlash avoidance is an appropriate choice of the required signs of the control forces. A methodology for the efficient determination of admissible signs for which preload is possible is part of ongoing work. A PPENDIX The special case of simply redundant actuation σ = 1, i.e. A1 ∈ R1,δ , is considered here. Denote with λk , σk , k = 1, . . . , δ the eigenvalues of AT1 A1 and I + AT1 A1 respectively. The matrix AT1 A1 ∈ Rδ,δ always has rank 1, so Pδ that λ1 6= T 0 and λk = 0, k ≥ 1. Because k=1 λk = trA1 A1 the only nonzero eigenvalue of 2 T AT1 A1 is λ1 = trA of P1 1 A1 T= kAk1 k . Thus the eigenvalues 2 T I + A1 A1 ≡ k=0 (A1 A1 ) are σ1 = 1 + kA1 k and σk = 1, k = 2, . . . , n. Hence I + AT1A1 is positive definite 2 with determinant det I + AT1 A1 = 1 + kA1 k . With 2(k−1) T k T (A1 A1 ) ≡kA1 k X A1 A1 the von-Neuman series −1 2 I + AT1 A1 = (−AT1 A1 )k yields, for kA1 k ≥ 0, k≥0
I + AT1 A1
−1
= I − AT1 A1
P
k≥0
k
(−1) kA1 k 2
= I − AT1 A1 /(1 + kA1 k ).
2k
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