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Desingularization of Flexible-Joint Parallel Robots

Parallel robots possess a characteristic type of singularities, called type II or drive singularities, inside their workspace. In the neighborhood of these singularities, the inverse dynamics solution grows unboundedly and the robot becomes uncontrollable. There is growing literature on methods that enable parallel robots to pass through drive singularities. Most of this literature relies on dynamic models that presume rigid joints. However, the flexibility of the drive train elements should also be taken into account for high accuracy. In this paper, we propose a systematic trajectory planning method for enabling flexible-joint parallel robots to pass through drive singular configurations. Our method generates admissible polynomial trajectories of degree eleven. Four conditions are derived and incorporated into the method to prevent undesired back-and-forth motion of the endpoint. This ensures not only an efficient operation of the robot but also the avoidance of unintended multiple occurrences of the same singularity. The boundedness of the inverse dynamics solution is also guaranteed.


Introduction
Parallel robots are used in a wide variety of applications and promise many more. For this reason, numerous studies have been devoted to the design and improvement of various parallel robots. Some recent works in this regard are [1][2][3][4]. However, "type II singularities" [5] or "drive singularities" [6] constitute a challenging issue in the use of parallel robots. The inverse dynamics solution is unbounded near these singularities, which results in saturation of the actuators and loss of control over the robot. Since such singularities lie generally in the interior of the workspace [5], a considerable amount of research has been conducted to develop trajectory planning methods for enabling parallel robots to pass through them with bounded actuator efforts. Ider [6] derived the "consistency conditions" in terms of the generalized accelerations as necessary conditions to be satisfied at the singular positions. Jui and Sun [7] introduced an approach that considers the bounds of the actuators. Briot and Arakelian [8] formulated an equivalent condition for consistency in terms of the forces and moments exerted on the moving platform. Özdemir [9] proved that although the consistency of the robot's dynamic equations at singular configurations is necessary, it is not sufficient to guarantee the boundedness of actuator efforts, and derived the additional necessary conditions in this respect by taking into account the time derivatives of the associated vanishing determinant at the instant of singularity. The conditions under which a planar 5R parallel robot experiences "high-order singularities" were derived in [10]. As remarked in [11], a trajectory planned to achieve the consistency of the dynamic model at the singularity may result in unintended back-and-forth motion of the endpoint. A method was proposed in [12] to prevent the possibility of such an undesirable motion.
All the above-mentioned studies on the trajectory planning of parallel robots are based on the assumption of nonredundant actuation. But when the effects of flexibility arising at the actuated joints due to the elasticity and structural damping of the drive train elements are significant, the robot should be treated as an underactuated system with doubled number of degrees of freedom [13,14]. Indeed, the issues related to the dynamics, trajectory planning, and control of such a system are quite different from those of its counterpart with rigid drive trains. Spong [13] proposed to model the elasticity of the joint as a linear torsional spring. Structural damping can be taken into account by adding a linear torsional damper in parallel to the model [15]. Although there are several studies [16][17][18] on the control and optimization of flexible-joint parallel robots, the motion is confined to be singularity-free in these studies and the literature is very limited on their trajectory planning for passing through drive singularities. Considering the torsional elasticity of the actuated joints, it was suggested in [19] that the degree of the trajectory polynomial must be at least twelve to pass through a singularity.
In the present paper, a novel trajectory planning method is proposed for desingularization of flexible-joint parallel robots, by which an admissible trajectory can be systematically generated using an eleventh-degree polynomial, that is, using a polynomial of a lower degree than that required in the related literature [19]. This degree reduction from twelve to eleven is particularly important since it also enables to solve the problem of avoiding any undesired reverse motion of the endpoint. Such a back-and-forth motion is obviously undesirable from an efficiency perspective. However, more importantly, it may -87 -lead to some additional unintended occurrences of the same singularity, as shown for the first time in the present paper. Although a singularity is removed at the intended time of passing through its corresponding configuration, the dynamic equations will be, in general, inconsistent at the times of its unplanned additional occurrences, yielding an unrealizable trajectory. Hence, such occurrences should be avoided, and when an eleventh-degree polynomial is used, the avoidance of reverse motion is guaranteed by ensuring the nonnegativity of a second-degree time polynomial over the task duration. The boundedness of the inverse dynamics solutions is also guaranteed by taking into account the first-order time derivative of the associated vanishing determinant, in accordance with [9,10].

Dynamic Modelling of Flexible-Joint Parallel Robots
Consider a rigid-link flexible-joint parallel robot with n rigid degrees of freedom and n motors at the actuated joints. The elasticity and structural damping of the coupling between the output shaft of the speed reducer of the i th motor and the corresponding driven link is modeled by a parallel arrangement of a linear torsional spring with a spring constant i k and a linear torsional damper with a damping coefficient i c [15]. Notice that there are n additional degrees of freedom due to the flexibility at the actuated joints. Let a i q represent the angular displacement of the link that is coupled to the output shaft of the speed reducer of the i th motor, the angular deflection of this coupling being It is convenient to study first the dynamics of the rigid-joint counterpart of the robot, where each motor (including its rotor and gears) is treated as a point mass fixed to the link on which the stator is attached [13,15] The mn  loop-closure constraint equations can be written in the form where j g 's are scalar functions. The equations of motion of the fully rigid parallel robot can then be expressed as  Mq N Q Aλ (5) where   is the vector of Lagrange multipliers, and A is   m m n  matrix whose elements are given by , 1, , , 1, , Now consider the flexible-joint parallel robot. The gear ratios i R are assumed to be large enough so that the rotational kinetic energy of each rotor is approximately due only to its own spin [13]. By this assumption, the coupling terms between the joint and actuator accelerations disappear and the equations of motion of the flexible-joint parallel robot can be obtained as [16,17]     aa      τ Jθ C θ q K θ q (7) ds    

Computation of Input Torques and their Boundedness near Drive Singularities
The inverse kinematic analysis of a flexible-joint robot is performed in the same manner as of its rigid-joint counterpart [16,20]. Given a task, assume that the inverse kinematic solution for it does not contain any singularities, and the time histories of the link variables 1 ,, m qq and their first-and second-order time derivatives are obtained for the prescribed vector of task variables Here t denotes the time variable. The inverse kinematic singularities of closed-loop chains are in general encountered at the boundaries of the workspace [5]. For this reason, they are not a major concern and are left out of the scope of this study. Now partition the matrices M and A and the vector N as -90 -Equation (8) can then be expressed as the following two equations: Having q , q and q obtained as explained above, unless the square submatrix u A is singular, the Lagrange multipliers can be computed by rearranging equation If all damping coefficients are nonzero, then matrix C is invertible and equation (14) is a system of n first-order differential equations for the variable θ of the form Thus, θ can be calculated by numerically integrating equation (17), and θ can be obtained by numerical differentiation. Finally, the time history of the vector τ can be computed from equation (7).
If some of the damping coefficients are negligibly small and taken as zero in the dynamic model, then matrix C is singular and equation (14) is a system of differential-algebraic equations for the variable θ . However, these equations are uncoupled since the off-diagonal elements of the matrices C and K are all zero. Therefore, a nonzero damping coefficient c  leads to a first-order differential equation of the form  (19) If all of the damping coefficients are assumed to be zero, then equation (14) is a system of algebraic equations whose solution is We present below also a generalized alternative method that can be used for any case of structural damping.
-91 -Writing equation (14) at time 1 v t  and rearranging yields By using the backward-difference method, we have Here, h is the time-step size. Substituting equation (23) into equation (21) and Drive singularities are the configurations where the matrix u A is not of full rank.
Near these configurations, the Lagrange multipliers, and hence the required input torques for realizing the prescribed task, are in general unbounded. In order to have a bounded inverse dynamics solution, the Lagrange multipliers should all have finite limits as the singularity is approached. Let it be assumed that u A is rank-deficient by one at the singularity time s t , which is generally the case [6,7]. Equation (16) can be rewritten in terms of the adjoint and determinant of When the robot is away from a drive singularity, the matrix u A and its adjoint matrix have full rank, and its determinant is nonzero. Therefore, the Lagrange multipliers can be regularly determined from equation (25) mn  of them are redundant. This is because the rank of the adjoint matrix of a square matrix P is 1 if P is rank-deficient by one [21]. Consequently, when u A loses one rank, all the individual equations in matrix equation (26) equivalently describe the same unique consistency condition. In the next section, we propose a novel method for planning a trajectory satisfying this condition. Our method also features two distinct tests: one for the nonzeroness of the first-order time derivative of the determinant of u A at the singularity time, and one for preventing undesired backand-forth motion of the endpoint. For the latter test, four additional conditions are derived, and in this manner, the possibility of unintended multiple occurrences of the same singularity is avoided.

A Novel Trajectory Planning Method
The previous section deals with the problem of removability of drive singularities of parallel robots in the presence of joint flexibility. It may be useful to note that drive singularities and the conditions under which they become removable would be identical for a flexible-joint parallel robot and its fully rigid counterpart since equation (15) does not contain any element of θ , or of its time derivatives. However, trajectory planning of flexible-joint parallel robots is quite different from and more complex than that of fully rigid parallel robots. This is because due to the elastic media at the transmission, the motors cannot instantaneously change the accelerations of the end-effector [15,16]. Motivated by this fact, in this section, we propose a new method of planning singularity-removed trajectories for flexible-joint parallel robots.
In the absence of obstacles, the optimal path between two points is the straight line joining these points. For this reason, we will focus here on tasks where the endpoint E of the robot has to perform a rest-to-rest motion along a straight line when the initial time is taken to be 0 t  . For a flexible-joint robot, the input torques instantaneously affect the snaps of the end-effector rather than its accelerations [15,16]. Hence, for the manipulator to be at rest at the start and end of the motion, the following eight conditions are required:  t t a a t a t a t a t a t The solution of the quadratic equation (44) is It is obvious that in order for the roots given by equation (48) to be real, the discriminant 1  must be nonnegative. Now, we propose the following method to obtain the coefficients of a rest-to-rest polynomial of the form of equation (37) that ensures the consistency of the robot's dynamic equations at a user-selected singularity crossing time: Step 1) Select a   sf 0, tt  such that 1 0  .
-96 - Step 3) If 1 0  , select and proceed with any one of the two distinct 11 a 's.
If in addition to the consistency, the first-order time derivative of   u det A is nonzero at s t , then the inverse dynamics solution is bounded as s tt  . Therefore, after the trajectory polynomial is obtained using the above method, this second condition should be checked to guarantee the boundedness of the required input torques near s t .
Another important consideration in trajectory planning is the prevention of any undesired back-and-forth motion of the endpoint [11]. This is important not only for the efficiency of the robot [12], but also for avoiding possible additional occurrences of the same singularity at times other than the selected s t . Such unintended occurrences make it, in general, impossible to realize the trajectory since at these times the dynamic equations will be, in general, inconsistent. It is a fact that if   where   bt is a second-degree polynomial of the form A term-by-term comparison of equations (50) and (51) yields

Case 1: A Consistent Trajectory Yielding a Determinant That Has Flatness 2 at the Singularity Time
In this first case, a consistent trajectory is planned by selecting s t equal to 0.5 s , which is half of the total task duration. This selection yields 1 0  . Therefore, there is only one real value for 11 a , which is 11 2520 a  . The remaining nonzero coefficients are then determined as: 5 1386 a  , 6 9240 a  , 7 25740 a  , -100 -

Case 2: A Consistent Trajectory That Results in Multiple Occurrences of the Same Singularity
To overcome the problem encountered in Case 1, s t is slightly changed to  . Thus, the inverse dynamics solution is bounded near this time. However, it is unbounded near the unintended singularity times 0.3668 s t  and 0.6328 s t  since the consistency condition is not fulfilled at these instants. The time histories of the required motor torques are shown in Figure 6.  Figure 6 Motor torques required in Case 2

Case 3: An Admissible Trajectory
It is seen in Case 2 that when s t is set to 0.5005 s , there are two distinct real solutions for 11 a . The consistent trajectory that is planned by choosing the first of them results in additional unintended occurrences of the same singular point.  Figure 8).

Conclusions
This paper contributes to the literature in the following ways:  We develop a systematic step-by-step trajectory planning method for flexible-joint parallel robots passing through drive singular configurations.  Our method generates a consistent trajectory with an eleventh-degree time polynomial, that is, with a time polynomial of a lower degree than that required in the related literature [19]. The importance of this -104 -achievement is that it enables transforming the problem of preventing unintentional reversal of the endpoint motion into the nonnegativity of a quadratic polynomial of time over the duration of the task.
 Although it is known that a trajectory, which is planned to be consistent at a drive singularity may result in undesired back-and-forth motion of the endpoint [11], we show here for the first time that this can lead to unintended multiple occurrences of that singularity, which should indeed be avoided.
 Four conditions are derived within the framework of the method to avoid undesired back-and-forth motion of the endpoint. Hence, the possibility of unintended multiple occurrences of the same singularity is prevented by the satisfaction of one of these four conditions.
 As previously shown in [9] for fully rigid parallel robots, we show in the present paper that the consistency requirement is also not sufficient for desingularization of flexible-joint parallel robots.
 The boundedness of the inverse dynamics solution is guaranteed by incorporating into the method a check for nonzeroness of the first time derivative of the vanishing determinant at the instant of singularity, in accordance with [9,10].
In order to show the effectiveness of our approach, we also present a detailed case study considering the planar 5R parallel robot with joint flexibility. In Case 1, the occurrence of a high-order drive singularity is demonstrated for the first time for a flexible-joint parallel robot. Case 2 illustrates, again for the first time, undesirable additional occurrences of the same singularity. Finally, all these problems are overcome in Case 3 by using our developed trajectory planning method.
The practical importance of this study is that it provides an efficient way of enabling the utilization of the entire workspace of flexible-joint parallel robots. In applications, this joint flexibility can be due to inevitable structural elasticity and damping of drive transmission elements or can be intentionally introduced for avoiding damage to the motors in case of unexpected impact loads. The method developed in this study can be employed in all these possible situations.