Proceedings of the 1999 IEEE International Workshop on Robot and Human Interaction Pisa, Italy - September 1999
Design of a New Tendon Driven Haptic Interface with Six Degrees of Freedom Antonio Frisoli, Fabio Salsedo, Massimo Bergamasco PERCRO Simultaneous Presence, Telepresence and Virtual Presence Scuola Superiore S. Anna Via Carducci,40,1-56127 PISA, Italy Email: ant
[email protected] Abstract
nected to the base by eight pistons, disposed in a triangulated pattern. By means of this static analogy, the mechanical architecture of the system recalls an octahedral-like geometry with two more linear actuators. But with respect t o the octahedral parallel manipulator classical designs [3], the mechanical system can be implemented by means of DC iron-less motors and steel cables as tendons, yielding an high fidelity forcefeedback desktop device.
A six degrees-of-freedom manipulandum with force feedback capabilitaes has been designed for using as an Haptic Interface (HI) in Virtual Environments and Teleoperators. The HI system is devised to address tasks performed by people with small tools, involving dextrous manipulation. The proposed manipulandum new kinematics is fully parallel and actuator redundant. Actuator redundancy refers to the addition of more actuators than strictly necessary to control the mechanism, without increasing the mobditg. The use of theS form of redundancy includes the ability to partial control the internal forces and augments the kinematic dexterity. A new tendon drive is used for each of the six degrees of freedom, allowing all actuators to be fized to the ground. Kinematic optimization of the dexterity and redundant actuation analysis of the manipulandum has been devel-
2
The kinematics of the legs of the HI is based on the closed &bar mechanism. An innovative tendon transmission has been devised to drive the closed &bar mechanism. It is composed of two tendons routed orderly over the pulleys mounted on each joint axis,as shown in figure 1. All the pulleys are idle, except the final driven pulleys of each tendon transmission, that are bolted t o the base link. The pulleys radii are the same for all the joints, but with different winding directions. So differently from classical tendon transmissions used in serial manipulators, the final driven pulley is grounded and it is not connected t o a moving driven link. This new tendon drive design allows, by properly choosing the tendon routing, to improve the kinematic performance of the closed &bar linkage, i.e. avoiding the singularities and improving the kinematic dexterity.
oped.
1
Description of the tendon driven leg kinematics
Introduction
Parallel manipulators have been srtensively studied for their favorable properties in terms of structural stiffness, position accuracy and good dynamic performance [5]. Their well known counterbalance is lack in workspace dimensions and complex direct kinematics law. Several parallel manipulators proposed in the literature are based on octahedral geometry kinematics [l]. With such a kinematics the legs form a “zigzag triangulated pattern” [l]that connect the base to the mobile platform. So the mechanical stiffness is increased as well as the singularities configurations are avoided. The new manipulandum that is presented in this paper, is composed by a mobile platform connected by four legs to the fixed platform. Two motors located on the base actuate each leg by a new tendon drive system. Since eight tendons are used t o control six degrees-offreedom the configuration of the tendon driven system according t o [4] is of type N+2. The tendon drive modifies the kinematic behavior of the system, so that it becomes statically equivalent t o a mobile platform con-
2.1
Kinematics of the planar tendon driven 5-bar closed linkage
We analyze now the properties of the tendon routing depicted in figure 1. Since the sum of the internal angles of a triangle is T , it is easy to show that for the angles of figure 1 the following differential relations hold:
Now since the tendon branch tangent t o two consecutive pulleys is constant independently from the close
0-7803-5841-4/99/$10.0061999IEEE
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Forces that the mechanism can /exen to the EF'
2
axis X
Case A Pulling Leg
Figure 1: Kinematics scheme of the tendon drive for the closed 5-bar linkage
Figure 2: Planar representation of the mechanisms equivalent to the tendon drive
2.2 5-bar posture, the displacement dui of the starting terminal of the tendon is determined only by the variations of the joint angles: dVi = R[(dp-tdB1) -de+ (del +d&+da) +db+ (dr+dO,)J (4) So by using the differential expressions (1,2,3) we obtain: dui = 4RdOi
(5)
Case B: Pushing Leg
Comparative analysis
We have studied the differential kinematics of the closed 5-bar linkage both with the direct drive of base joints and with the new tendon drive, in order to point out the difference in kinematics performance. Kinematics performance have been compared computing over all the workspace the manipulability ellipses of the two driving systems. The results of an exemplifying case study are reported in Figures 3 and 4.
The above equation is very meaningful. Since
Z1 = 21 cos(O1)
3
l2 = 22 cos(02)
+
dZl = -21 sin(&) del d12 = -21 sin(02)de2
7
(6) (7)
by the duality principle between statics and kinematics, the action of the two tendon tensions TIand T2 is equivalent to two linear actuators directed along QP and R P with thrusts:
F' = TI4R FZ = T2-21 sin(&) 4R 21 sin(Ol)
(8)
:LA 0 0
But since tendon can generate only. tension forces, the previous static analogy is incomplete and, depending by the implemented routing, the equivalent pistons can either only pushing upwards or pulling downwards. So the kinetestatic behavior of a tendon driven closed 5-bar linkage can be reduced to one of the equivalent mechanisms shown in figure 2. This mechanical analogy is very worthwhile, since it permits to explain clearly the force capability of the tendon driven 5-bar linkage. The forces applied at the End Effector (EF) must be comprised in the angle formed by the two equivalent thrust vectors QP and R P , with the sign determined by the routing.
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1W
M,
Figure 3: Manipulability ellipses without the tendon drive The manipulability ellipses far the tendon driven 5bars mechanism have a rounder shape than those of the 5-bars mechanism actuated at the joints. So the proposed driving system improves the kinematics isotropy of the mechanism. The manipulability,i.e. the ellipses area, is also greater in the tendon driven mechanism.
3 04
ruling the statics of the HI is the dual of the Jacobian equation:
( ;) = JLt
50
---
with F and r being the external force and torque on the moving platform and t being the eight-dimensional vector of tendon tensions. The HI can exert forces and torques of arbitrary directions if and only if the kernel of Jt",tcontains a vector t whose components are all positive. The points of the workspace where such a condition is verified belong to the controllable workspace. Our aim is to enlarge the controllable workspace to the kinematically reachable workspace of the mechanism. So we have studied all the possible symmetric spatial arrangements of four legs t o find the most suitable architecture for an HI design that maximizes the controllable workspace. Finally we have opted for the following architecture: the legs are located with an axial symmetry of 90" around an axis normal to the base plane; the base axes of the legs lay in the base plane; both the pushing and the pulling legs are two; the pulling and pushing legs are placed in alternate way around the symmetry axes.
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Figure 4: Manipulability ellipses with the tendon drive
2.3
(9)
Extension to six degrees-of-freedom kinematics
The mechanical designs of both closed 5-bar linkages with a pushing type drive and with a pulling type drive have been developed. Then these two mechanisms have been assembled with a ball joint and with a rotational joint, as shown in figure (5) , to give raise to two type of six-degree of freedom kinematic components, later on called simply pushing and pulling legs.
Figure 6: CAD parametric model of the developed HI A CAD parametric model of the HI has been designed and is shown in figure 6.
Figure 5: CAD model of a pushing leg
3
Then four legs have been assembled with a mobile platform and a fixed platform in a six degree-of-freedom parallel manipulator. Such a parallel manipulator is actuators redundant since eight command variables, namely eight tendon tensions or displacements, are independently used t o control six degrees of freedom. On the other side, the constraint on the positive sign of the tendon tension [4], limits the actuation capability of the HI. The equation
Formal definition of the HI Kinematics
In this section the kinematic parameters of the architecture are defined and so the frames of reference necessary for the analysis of its motion. In Figure 3, a frame of reference Oxye integral with the base platform, with the z axis coincident with the HI symmetry axis, is used to define the absolute movements of the moving parts of the HI.
305
mobile reference frame
groundpla@mn
grvund reference frame
Figure 7: The base (Oxyz)and moving (R(C7) frames of reference and the base links (QiRi)and end-point (PI) of the legs A frame of reference R(Cq integral with the moving platform is used to specify the EF position of the HI. The transformation matrix T between the n(Cq and Oxyz is parameterized by six parameters R,, R,, R, ,&, e,, 8, in the following way:
T=
[
1
with Re, and Re: the orthogonal matrixes associated respectively to the rotations 8, and 8, and
i=
[?]
The vector Hi is the distance between the origin of the k e d and the moving frame when the HI is in the central position. The position of the end tool frame of reference flccq relative to the base frame Oxyz is specified by the displacements R,, fl,, R, of the moving frame from its initial position. The orientation of R(
