SPAEA Modeling Stewart Platform Errors

Nominal Model

A Stewart Platform consists of six variable length links connecting a stationary base to a movable platform. The legs can be connected to the base and platform by either ball joints and/or universal (Hooke) joints (sometimes with an additional pivot). For the general Stewart Platform, the base and the platform can be any shape whatsoever as long as the resulting structure remains stable. For example, a structure with regular hexagons for both the base and the platform would result in an unstable configuration. By changing the lengths of the legs, the position and orientation of the platform can be changed with respect to the base, thus causing the relative motion of the platform. In order to understand the relationships of the general Stewart Platform, a kinematic model is needed.

Preliminaries and Definitions

The nominal model of the Stewart Platform is based on the following assumptions:

• The ball joints and the universal joints (and pivots, if used) are perfect. They have exact centers about which perfect rotation can occur.
• The actuators in the legs are perfect. They move with only one degree of freedom which passes through the joint centers, and the length can be read from one joint to the other without offset.
• The base and platform structures are accurate so that the position of the joint centers are precisely known.

Figure 1 Nominal Model of a Stewart Platform and Figure 2 Vector Chain for Leg i will help illustrate some definitions. A base coordinate frame {B} is arbitrarily embedded in the base, and likewise a platform coordinate frame {P} is placed on the platform. Vectors referenced in coordinate frame {B} are denoted by ^Bv , while vectors referenced in {P} are denoted ^Pv. Additional coordinate frames may be used to describe relationships between the base frame and a world frame, or between the platform frame and the tool frame, but these other frames are omitted here for clarity and with no loss of generality.

Figure 1 Nominal Model of a Stewart Platform	Figure 2 Vector Chain for Leg i

The joints on the base are denoted by B_i, i = 1, 2..6, and the position of the joint centers with respect to {B} are given by the vectors . Similarly, the joints on the platform are denoted by P_i, and the position of the joint centers with respect to {P} are given by vectors . The six legs are denoted by L_i, with vector from B_i to P_i defined by , where is the magnitude (the length of the leg), and l_i is a direction unit vector. Each leg has some sort of transducer that is able to read the length of the leg, so that is a measurable quantity. A vector will be used to describe the measured leg lengths.

The two coordinate frames {B} and {P} can be related to each other through a displacement ^Bq=[x y z]^T and an orientation matrix ^BR_P.

	(1)

The vector ^Bq gives the relative displacement of the origin of {P} from the origin of {B} in terms of the its longitudinal (x), lateral (y), and vertical (z) components. The orientation matrix ^BR_P is composed of direction cosines defined such that a unit vector along x_P has components (I_x I_y I_z) in the base frame {B}. Similarly, a unit vector along y_P has components (J_x J_y J_z) in {B}, and a unit vector along z_P has components (K_x K_y K_z) in {B}. Of the nine variable in the orientation matrix, six are redundant since only three values determine the orientation. Of the many ways to describe orientation using three parameters, the most widely used is the yaw-pitch-roll angles, , , and (sometimes called a Euler zyx rotation set). The orientation of frame {P} is obtained by first rotating frame {B} an angle (roll) about the z axis, then rotating the resulting frame an angle (pitch) about the rolled y axis, and finally, rotating the resulting frame an angle (yaw) about the rolled and pitched x axis. The orientation of frame {P} can be also described by the equivalent rotation set of first rotating frame {B} an angle (yaw) about the x_B axis, then rotating the resulting frame an angle (pitch) about the y_B axis, and finally, rotating that frame an angle (roll) about the z_B axis (see Paul for a further explanation). Thus, an orientation vector will be defined using the yaw-pitch-roll angles as so that the orientation matrix is given by

	(2)

The pose will be formalized by defining a pose vector P=[x y z ]. The pose vector thus defines the position of the platform with respect to the platform [by its longitudinal (x), lateral (y), and vertical (z) displacements], and its orientation [by the yaw (), pitch (), and roll () angles].

For robust operation and control of a Stewart Platform, it is usually necessary to know both the pose of the platform P, and the lengths of the legs L.

Inverse Kinematics

The inverse kinematics problem of the Stewart Platform deals with calculating the leg lengths when the pose is given. In effect, it is a mapping from global pose to local actuator lengths, P to L. With the assumptions of the nominal model, the inverse kinematics of a Stewart Platform are uncomplicated, yielding a nonlinear closed form solution.

For convenience, let ^Bu_i be a representation of a vector from the origin of {P} to P_i, but referenced in the base frame so that

	(3)

Then, following the vector chain in Figure 1 from {B} to {P} to P_i to B_i back to {B} yields

	(4)

which rearranges to

	(5)

The length of leg i can then be determined by taking the magnitude of Equation 5.

	(6)

Expanding Equation 6 and squaring both sides for ease of reading gives

	(7)

Equations 6 and 7 present the inverse kinematics for the nominal representation of a general Stewart Platform. By specifying the position and orientation of the platform, P, along with knowledge of the kinematic parameters, the leg lengths, L, can be calculated.

Accurate Model

No Stewart Platform is a perfect situation generator. In fact, the position and motion of the tool will inevitably vary from nominal. Among the most likely phenomena to introduce errors in pose are the following [Liegeois]:

• Faults in manufacture/assembly: For example, the joints will not be in their ideal locations.
• Kinematic errors in the actuators: The leg actuators will not only move in one degree of freedom, but will move in six degrees of freedom. Also, problems such as backlash will be present.
• Elastic deformation of the axes, segments, motor parts, and transmission devices.
• Variations of the kinematics resulting from wear.
• Poor repeatability.
• Deformations caused by temperature change.
• Errors caused by the control system: precision and linearity of the position sensors, resolution of the servo systems, numerical precision of the controller, etc.

The direct relationship between these error sources and resulting pose errors is not well known. Therefore, the effects of the error sources are incorporated into error vectors that help create an accurate model of the Stewart. This model should shed light on the major contributors to pose inaccuracies and aid in future design and manufacture of Stewart Platforms.

A paper outlining the error model has been submitted to the CIRP. Until publication, however, the accurate model used in the software will not be presented here (please check back at mid March, if interested). Information on the accurate model can be obtained through email.