Parallel manipulators have received wide attention in recent years. Their parallel structures offer better load carrying capacity and more precise positioning capability of the end-effector compared to open chain manipulators. In addition, since the actuators can be placed closer to the base or on the base itself the structure can be built lightweight leading to faster systems (Gunawardana & Ghorbel, 1997; Merlet, 1999; Gao et al., 2002 ).
It is known that at
The previous studies related to the drive singularities mostly aim at finding only the locations of the singular positions for the purpose of avoiding them in the motion planning stage (Sefrioui & Gosselin, 1995; Daniali et al, 1995; Alici, 2000; Ji, 2003; DiGregorio, 2001; St-Onge & Gosselin, 2000). However unlike the kinematic singularities that occur at workspace boundaries, drive singularities occur inside the workspace and avoiding them limits the motion in the workspace. Therefore, methods by which the manipulator can move through the drive singular positions in a stable fashion are necessary.
This chapter deals with developing a methodology for the inverse dynamics of parallel manipulators in the presence of drive singularities. To this end, the conditions that should be satisfied for the consistency of the dynamic equations at the singular positions are derived. For the trajectory of the end-effector to be realizable by the actuators it should be designed to satisfy the
2. Inverse dynamics and singular positions
Consider an n degree of freedom parallel robot. Let the system be converted into an open-tree structure by disconnecting a sufficient number of unactuated joints. Let the degree of freedom of the open-tree system be
and can be expressed at velocity level as
Equation (3) can be written at velocity level as
The dynamic equations of the parallel manipulator can be written as
where M is the
Combining the terms involving the unknown forces
The inverse dynamic solution of the system involves first finding
For the prescribed x(t),
Singularities may also occur while solving for the actuator forces in the dynamic equation (8), when
In the literature the singular positions of parallel manipulators are mostly determined using the kinematic expression between
References (Sefrioui & Gosselin, 1995 ; Daniali et al, 1995; Ji, 2003) name the condition
3. Consistency conditions and modified equations
At the motion planning stage one usually tries to avoid singular positions. This is not difficult as far as inverse kinematic singularities are concerned because they usually occur at the workspace boundaries (DiGregorio, 2001). In this paper it is assumed that
3.1. Consistency conditions and modified equations when rank(A) becomes m-1
At a drive singularity, usually rank of A becomes
Equation (15) represents the
Now, because equation (13) holds at the singular position, there exists a neighborhood in which the first term in equation (16) is negligible compared to the other terms. Therefore in that neighborhood this term can be dropped to yield
3.2. Consistency conditions and modified equations when rank(A) becomes r<m
In the general case where the rank of
The consistency relations are obtained as below
3.3. Inverse dynamics algorithm in the presence of drive singularities
When the linearly dependent dynamic equations in equation (8) are replaced by the modified equations, equation (8) takes the following form, which is valid in the vicinity of the singular configurations.
where in the case the
In the general case when the rank of
Find the loci of the positions where the actuation singularities occur and find the linear dependency coefficients associated with the singular positions.
If the assigned path of the end-effector passes through singular positions, design the trajectory so as to satisfy the consistency conditions at the singular positions.
, and from kinematic equations.
If the manipulator is in the vicinity of a singular position, i.e.
where is the singularity condition and is a specified small number, calculate from eqn (27) and then find (hence T) from equation (22).
If the manipulator is not in the vicinity of a singular position, i.e.
, find (hence T) from equation (8).
. If the final time is reached, stop. Otherwise continue from step 3.
4. Case studies
4.1. Two degree of freedom 2-RRR planar parallel manipulator
The planar parallel manipulator shown in Figure 1 has 2 degrees of freedom (
The moving links are uniform bars. The fixed dimensions are labelled as
The loop closure constraint equations at velocity level are
Then the task equations at velocity level are
The mass matrix M and the vector of the Coriolis, centrifugal and gravitational forces R are
Since the variables of the actuated joints are
Then the coefficient matrix of the constraint and actuator forces,
The drive singularities are found from
When the end point comes to
Hence the time trajectory
An arbitrary trajectory that does not satisfy the consistency condition is not realizable. This is illustrated by considering an arbitrary third order polynomial for
For the time function s(
Furthermore, even when the consistency condition is satisfied,
The coefficients of the constraint forces in eqn (38) are
which in general do not vanish at the singular position if the system is in motion.
Once the trajectory is chosen as above such that it renders the dynamic equations to be consistent at the singular position, the corresponding
4.2. Three degree of freedom 2-RPR planar parallel manipulator
The velocity level loop closure constraint equations are
The prescribed position and orientation of the moving platform,
Let the joints whose variables are
The link dimensions and mass properties are arbitrarily chosen as follows. The link lengths are
The generalized mass matrix M and the generalized inertia forces involving the second order velocity terms R are
For the set of actuators considered, the actuator direction matrix Z is
Hence, drive singularities occur inside the workspace and avoiding them limits the motion in the workspace. Avoiding singular positions where
The desired trajectory should be chosen in such a way that at the singular position the generalized accelerations should satisfy the consistency condition.
If an arbitrary trajectory that does not satisfy the consistency condition is specified, then such a trajectory is not realizable. The actuator forces grow without bounds as the singular position is approached and become infinitely large at the singular position. This is illustrated by using an arbitrary third order polynomial for
For the time function
Bad choices for
However, even when the equations are consistent, in the neighborhood of the singular positions
Once the trajectory is specified, the corresponding
A general method for the inverse dynamic solution of parallel manipulators in the presence of drive singularities is developed. It is shown that at the drive singularities, the actuator forces cannot influence the end-effector accelerations instantaneously in certain directions. Hence the end-effector trajectory should be chosen to satisfy the consistency of the dynamic equations when the coefficient matrix of the drive and constraint forces, A becomes singular. The satisfaction of the consistency conditions makes the trajectory to be realizable by the actuators of the manipulator, hence avoids the divergence of the actuator forces.
To avoid the problems related to the ill-condition of the force coefficient matrix, A in the neighborhood of the drive singularities, a modification of the dynamic equations is made using higher order derivative information. Deletion of the linearly dependent equation in that neighborhood would cause task violations due to the removal of a task. For this reason the modified equation is used to replace the dependent equation yielding a full rank force coefficient matrix.
The elements of M and
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