International Journal of Advanced Robotic Systems Kinematically Redundant Parallel Haptic Device with Large Workspace Regular Paper

In this paper, a kinematically redundant parallel haptic device with large workspace is presented. The haptic device has a similar kinematic structure to the well‐ known Delta manipulator. However, it has a special arrangement of actuators and one redundant actuator added to the third leg. The proposed haptic device has essentially 4‐DOF, however, only three translational DOF are used for 3‐DOF positioning and force reflection, and one rotational DOF by the redundant actuator is used to increase well‐conditioned workspace. If the redundant actuator's angle is controlled to follow x and y position, the haptic device has a large cylindrical workspace and can maintain good kinematic and statics performance over the whole workspace. The kinematics and workspace are analysed, and the optimal design method of finding minimum link lengths to satisfy prescribed workspace is presented. Finally, the prototype haptic device and control experiment result are presented.


Introduction
Haptic devices have been extensively investigated and applied to many fields, such as teleoperation, robotic surgery, bioengineering, computer-aided design, etc.In order to provide realistic force reflection and high manipulability to an operator, the mechanism of a haptic device should have low inertia, high stiffness, large force reflection capability, good kinematic conditioning, as well as large workspace, back-drivability, low friction and small moving inertia.
In general, serial-kinematic haptic devices [1][2][3][4][5][6][7][8] have large workspace, but relatively small force reflection and low stiffness.In order to further increase force reflection and stiffness, parallel-kinematic manipulators have been employed as haptic device mechanisms [9][10][11][12][13][14].One of the most famous parallel-type haptic devices may be Delta haptics [15][16][17][18].The Delta parallel manipulator has large force reflection and high stiffness due to parallel structure and parallelogram.However, it has a smaller workspace compared to the serial one.A new parallel manipulator similar to Delta called "Tau" with a large cylindrical workspace as with a SCARA robot is also presented [19][20][21].However, the Tau manipulator consists of three different leg configurations and the kinematic performance is not symmetric.Comprehensive reviews of the mechanisms of haptic devices can be found in [22,23].In general, redundancy can improve the workspace and performance of a parallel manipulator [24,25].By adding more than one active joint to a non-redundant parallel manipulator, a kinematically redundant parallel manipulator can be constructed.
In this paper, a kinematically redundant parallel manipulator for a haptic device is conceived, which retains the advantages of both serial (large workspace) and parallel (large force and high stiffness) manipulators.The proposed parallel manipulator has the special arrangement of actuators and one redundant actuator for large workspace and good performance.The position, Jacobian and workspace analyses are performed.The optimal design to find minimum link lengths satisfying the prescribed workspace is presented.Finally, a numerical example of the optimal design, prototype haptic device and control experiment result are presented.

Position Analysis
As shown in Fig. 1, the kinematic structure of the proposed haptic device is similar to that of the wellknown Delta manipulator with three R-Pa (Revolutespatial Parallelogram) legs [15].However, the moving platform of the proposed haptic device is connected to the fixed base by two R-Pa legs on the horizontal plane and one R-R-Pa leg on the vertical plane, where R denotes an actuated revolute joint.Specifically, the axes of the first and second rotary actuators (R1 and R2) are placed along the z-axis and the axis of the third rotary actuator (R3) is perpendicular to the z-axis.One redundant actuator (R4) rotates the third rotary actuator and leg about the z-axis together with the angle of  .This parallel haptic device has 4-DOF, i.e., three translations and one rotation about the z-axis ().It is clear that the angle of the moving platform is equal to that of the redundant actuator's angle () due to the parallelogram of 3,2 l .
If the redundant actuator is controlled to follow input x and y position, i.e., Atan2( , ) , the connecting link of the third leg ( 3,2 l ) can be kept vertically and then 3,3 0   .With this control method, the workspace is not limited by the range of 3,3  and the manipulator can maintain better kinematic and statics performance over the whole workspace than that of 3,3 0   (refer to Fig. 3).The parallel haptic device can be treated as a 3-DOF positioning one.The first two actuators determine the x and y positions of the moving platform, and the third actuator controls the z position.If the first two actuators and the redundant actuator can make a complete circle around the z-axis, the whole workspace becomes a large hollow cylinder.
As shown in Fig. 1, the global reference frame is attached to the fixed base at O , and the ith local frame is attached at point i A .It is noted that the frames 1 A and 2 A coincide with the reference frame.
For each leg, the following vector-loop equation should be satisfied [28].
Expressing Eq. ( 1) with respect to each local frame i A where , i i j v is the jth vector in the ith leg expressed in frame i A and , i j s is the jth scalar quantity in the ith leg in this paper.The unit directional vectors of links are given by ,1 ,1 ,1 where . The position vector of , is obtained by In the following analyses, it is assumed that the redundant actuator's angle  is given and the 4-DOF parallel manipulator will be considered as a 3-DOF positioning one.The inverse position analysis can be defined as the problem of finding the joint angle vector, for given the end-effector position vector, [ , , ] T x y z p p p  x .First, 3,i  can be calculated from the third element of Eq. (2) as By summing the squares of ,  2), an equation with only ,2 i  can be obtained.
Solving Eq. ( 6) for ,2 where   , the given position is out of the workspace.
The problem of the forward position analysis is to find end-effector position x for the given joint angle vector q .Rewriting Eq. ( 2) yields

')
By summing the squares of the three components of Eq. (2'), the following sphere equation for the ith leg is obtained by The plane equation that contains the circle of intersection made by the spheres of leg 3 and leg j is determined by subtracting Eq. ( 9) for i j  from Eq. ( 9) for 3 i  : ) ) ) Solving Eq. ( 10) for x p and y p in terms of z p and then substituting the resulting expressions into Eq.( 9) for where q e e e e   .
Once z p is found, x p and y p can be determined by back substitution into Eq.(10).

Jacobian analysis
Differentiating Eq. ( 2) with respect to time yields where i x  is the linear velocity of the moving platform and ω is angular velocity of the jth link of the ith leg with respect to frame i A .To eliminate the passive joint rate, dot-multiplying both sides of Eq. ( 12) by Expressing the vectors in Eq. ( 13) in frame i A gives ,1 ,1 0 0 and ,2 i i u is given in Eq. ( 3).Writing Eq. ( 13) three times, once for each for i=1, 2, and 3, yields three scalar equations, which can be assembled in matrix form as where If q J is not singular, Eq. ( 15) can be rewritten by where Using the principle of virtual work, the statics relation is obtained by denote the endeffector force and joint torque vectors, respectively.The stiffness mapping in the joint space can be expressed by where denotes a 3 3  diagonal matrix representing the joint stiffness.Applying the velocity and statics relations to Eq. ( 19), the stiffness mapping in the Cartesian space can be expressed by where and the stiffness matrix is given by

Workspace analysis and optimal design
If the redundant actuator is controlled simply with 1 tan ( / ) , the kinematically redundant parallel haptic device can have a large well-conditioned workspace.Furthermore, using the control method, the design problem can be reduced to a planar one.In other words, this manipulator is symmetric at every x z  plane with the incident angle of  to the xz plane.The optimal design will be performed on the x z  plane.
Figures 3(a) and 3(b) demonstrate that the kinematic and statics performances (isotropy condition [26] and minimum force transmission capability [27]) when using this control method are larger than those when using a fixed angle, 0   , which is the case for a non-redundant manipulator such as the typical Delta.The isotropy condition and minimum force transmission capability are defined by First, the minimum length of b can be selected to prevent the interference of parallelograms.The length of b is dependent on the width of a parallelogram.It is noted that a smaller length of (a -b) is desirable for smaller link lengths of leg 3 (refer to Fig. 5 and Eq.(30b)).In the prototype development, a=85 and b=35 [mm] are chosen.
Once [a, b] are determined, the objective of the optimal design can be defined as a problem to determine a set of minimum link lengths to satisfy the prescribed work area.
The kinematic design variables of the optimal design are selected as [ , , , ] where The prescribed work area on the x z  plane is defined as a rectangle.
The angular limits of the passive joints are chosen by where the angular limit of the second passive joint is introduced to prevent serial singular configuration and that of the third passive joint comes from a spherical joint limit.
First, the height of the work area is limited by the second link lengths and the third joint angle limits of the first and second legs as The next step is to determine 1 p l by considering the minimum and maximum configurations of the first and second legs.For simplicity of expression, the leg number is omitted.As shown in Fig. 4, the min x of the endeffector should be smaller than 5 P .
max max sin sin( ) As shown in Fig. 5, the max x of the end-effector should be larger than 1 P or 3 P .
The minimum length of the arms ( 1 p l ) of the first and second legs can be determined by The numerical iteration method is used to determine In general, the work area including the prescribed one is determined by the intersection area of the four circles and / 2 / 2 h z h    .In Fig. 6, the centres and radii related to the first and second legs are given by  in Eq. (30a).The resulting intersection is obtained as shown in Fig. 7.In this design, the minimum ) is selected for larger force transmission capability at the end-effector.Table 1 shows the optimal design result.The kinematic parameters in the prototype are slightly different from the optimal design result.
The workspace of the prototype haptic device is shown in Fig. 8, where the following rotary actuator ranges from the mechanical interference are considered.It has been demonstrated that the workspace of the haptic device with a redundant actuator is much larger than that without a redundant actuator ( 0   ).
Figure 9 shows the force transmission capabilities along the x-, y-and z-axes for the maximum actuator torque,

Conclusion
In this paper, a Delta-type parallel manipulator with one redundant actuator is conceived for the 3-DOF positioning haptic device with a large workspace.The position, Jacobian and workspace analyses of the 3-DOF positioning parallel haptic device are presented.The optimal design method of finding minimum link lengths to satisfy the prescribed workspace is developed and applied to the prototype haptic device design.The prototype haptic device and 4-axis PC-based real-time controller using xPC target have been developed.Finally, it is demonstrated through analysis and experiment that by adding one redundant actuator to the third leg and using the simple control method of , the workspace can be increased significantly and good kinematic performance can be maintained over the whole workspace.

Figure 4 .
Figure 4. Work area limited by legs 1 and 2.

5 . 2 220
Prototype development and control experimentIn the prototype design, the prescribed work area and other kinematic parameters are selected as the intersection of Eqs.(30a) and (30b) does not exist for the given angular limits, max 

Figure 7 . 1 z
Figure 7.The intersection area of 1 z l and 2 z l .

Figure 8 .
Figure 8. Workspace of the prototype haptic device.

Figure 9 .
Figure 9. Mesh plot of force transmission capability on the x z plane.

Figure 10 .Figure 11 .
Figure 10.Mesh plot of Cartesian stiffness on the x z  plane

Figure 12 .
Figure 12.System configuration of a haptic device.

Figure 14 .
Figure 14.Control experiment on the redundant actuator when the moving platform moves along the y-axis.

Table 1 .
[29]1.11[Nm]As shown in Figs.11 and 12, the 4-axis controller consists of Host PC with Simulink and Target PC with xPC Target from MathWorks.Figure13shows the prototype haptic device which was upgraded from the first prototype in[29].Three DC servo motors and wire-driven gears (gear ratio=9.5:1)are used for 3-axis force reflection and joint angle sensing.The gravity force of the moving parts is compensated by the DC servo motors.One geared AC servo motor (gear ratio=45:1) is used for the position control of  .Optimized design parameters.