The Maximization of a Serial System S Reliability

DYNAMICS OF THE STEWART PLATFORM

J. Wittenburg

University of Karlsruhe / Germany

A Stewart platform is a rigid body supported by six telescopic legs the ends of which are spherical joints Pi and Qi fixed on the platform and on the supporting frame, respectively (see Fig.l). In an inertial cartesian base the articulation points Qihave position vectors (i — 1,...,6) and in a platform-fixed base the articulation points Pi have position vectors (i = 1,..., 6). Neither the six points on the platform nor those on the frame need be coplanar or otherwise regularly arranged.

In a general position five legs of given constant lengths determine a linear complex which gives the platform a single degree of freedom, namely a screw motion about the axis of the linear complex. A sixth leg of constant length which is not a complex line of this linear complex connects the platform rigidly to the frame. If the leg lengths are then made variable the degree of freedom of the platform is six. In addition, each leg has the degree of freedom of rotation about its longitudinal axis. This degree of freedom is without interest. In practice it is often suppressed by replacing one spherical joint per leg by a universal joint. Stewart platforms find applications whereever the six position variables of a body must be controlable both fast and with high accuracy. Examples are grippers of robots, carriers of cameras or antennas, carriers of vehicles in drive and flight simulators and in tool machines. Stewart platforms are also found in the field of vibration control and in micro system' technology.

The aim of this paper is to formulate for the system composed of platform and legs six dynamics equations of motion. As origin of the platform-fixed base the center of mass C is chosen, and the axes are directed along the central principal axes of inertia. The position and the angular orientation of in are described by the three coordinates x1,x2,x3of the position vector of C in the base and by Euler-Rodrigues parameters They form the column matrices und (matrices are denoted by underscore; the exponent T stands for transposition). In the case the base coincides with .The coordinate transformation matrix defined through the equation is

(1)

Let be the absolute angular velocity of the platform and letbe the column matrix of its coordinates in the principal axes frame.The parameters are the solutions of the kinematic differential equations (is the coordinate form of the vector product.

.(2)

The dynamics equations to be developed are formulated in terms of the variables, and - Starting point is the principle of virtual power. For a general system of n rigid bodies labeled i=1,..., n it has the form

(3)

(position vector of the body icenter of mass, absolute angular velocity , mass mi, inertia tensor Ji, vertical unit vector , resultant force other than weight, resultant torque about the body icenter of mass). In the present case n = 13 (platform plus two segments of each of the six legs). The quantities related to the platform are identified by omitting the index i. The coordinate form of this contribution is

(4)

The contribution of a single leg is developed from Fig.2. It depicts the platform, the bases and, the position vector of C and the articulation points Q and P (position vectors fixed in and fixed in ). The leg is composed of the segments 1 and 2 with centers of mass C1 and C2, respectively. The positions of these points are specified by the constant parameters l1 and l2. Let L be the variable leg length and let be the variable unit vector pointing from Q to P. From the figure it is seen that

(5)

Each of the segments 1 and 2 is assumed to be an infinitesimally thin rod with zero principal moment of inertia about the longitudinal axis and with two identical principal moments of inertia about lateral axes through the center of mass. This has the consequence that angular velocity and angular acceleration components along the longitudinal axis do not appear in Eq.(3) (if the leg has spherical joints at both ends P and Q then the said longitudinal components can actually be identically zero). With these assumptions both segments have identical absolute angular velocities orthogonal to the leg axis. In what follows this angular velocity is called. The angular accelerations of the two segments are also identical and orthogonal to. The expression inEq. (3) in which J represents an inertia tensor reduces to where J is now the principal moment of inertia about the lateral axis. At the centers of mass forces and, respectively, are acting. Here, represents the hydraulic control force. The virtual power of is. Here, is the virtual velocity increment of the platform-fixed point P (the points C1 and P have identical velocity components along).

The segment 2 has the fixed point Q. The expression in Eq. (3) has a simpler form if as reference point this point Q is chosen instead of the center of mass C2. Then,. With the principal moment of inertia J2 about the lateral axis through Q and with the torque about Q the contribution of segment 2 to the sum is the expression. Hence, the contribution of the entire leg is

(6)

I hisis already the coordinate formulation. The column matrices contain the coordinates of therespective vectors in the base.

In what follows the vectors and are expressed in terms of kinematical quantities of the platform. The platform-fixed point P has the position vector, the velocity and the acceleration:

(7)

From the second equation it follows that

(8)

The vector is also,whence follows

(9)

Dot and cross multiplications of this equation with yield the formulas

(10)

Time differentiation of produces the expression

(11)

and after substitution of the expressions for , and

(12)

Fig.2 shows that. The second time derivative is or with the expressions for, and

(13)

where T is the symmetric tensor

(14)

(unit tensor I). From this it follows that

(15)

This ends the kinematics analysis.

Next, the coordinate matrices of the vectors needed in (6) are formulated. The vectors and have in the platform-fixed base coordinate matrices and , respectively. The coordinate matrices in are and respectively, where is the transpose of the matrix in Eq.(l). The vector has in the coordinate matrix with.With the coordinates of the unit vector is calculated. Let be its coordinate matrix in .Then, the tensor T has the coordinate matrix . In terms of these expressions the coordinate matrices of the vectors , , , and have the forms

(16)

For the sake of simplicity lower-order derivative terms are indicated by dots. It is a simple matter to write down also these terms. These expressions arc substituted into (G). In multiplying out the symmetric (3 x 3)-mass matrix.

(17)

is encountered repeatedly. The final result for the contribution of a single leg is found to be

(18)

Again, dots indicate lower-order derivative terms.

Each of the six legs contributes, with different parameters and, such an expression. Let and (i = 1...6) be the quantities for the leg i. The contribution of the platform which is given in (4) has the same general form with simpler matrices though. According to Eq.(3) the sum of all seven expressions equals zero. This equation has the form. Theexpressions indicated by dots must both be zero. This yields the desired equations of motion. They have the form

(19)

where the individual matrices are (is the diagonal matrix of the principal moments of inertiaof the platform)

(20)

In the (6 x 6)-matrix Kcolumn i (i = 1, ... 6) is composed of the column matrices and , stacked one on top of the other. The matrices and Kdepend on the variables,.The Eqs.(19) have to be integrated together with the kinematics Eqs.(2).

Figure1:Stewart platform

Figure 2: Platform and a single leg with reference bases, position vectors and forces