1 Definition of the Stewart Platform Geometry
++Stewart platforms are generated in multiple steps: +
+-
+
- Definition of the frames +
- Definition of the location of the joints +
- Computation of the length and orientation of the struts +
- Choice of the rest position of the mobile platform +
+This steps are detailed below. +
+1.1 Frames Definition
++We define 4 important frames (see Figure 1):
- \(\{F\}\): Frame fixed to the Fixed base and located at the center of its bottom surface. This is used to fix the Stewart platform to some support.
- \(\{M\}\): Frame fixed to the Moving platform and located at the center of its top surface. This is used to place things on top of the Stewart platform. -
- \(\{A\}\): Frame fixed to the fixed base. -It defined the center of rotation of the moving platform. -
- \(\{B\}\): Frame fixed to the moving platform. -The motion of the moving platforms and forces applied to it are defined with respect to this frame \(\{B\}\). +
- \(\{A\}\): Frame fixed to the fixed base. +
- \(\{B\}\): Frame fixed to the moving platform.
-Then, we define the location of the spherical joints: +Even though frames \(\{A\}\) and \(\{B\}\) don’t usually correspond to physical elements, they are of primary importance. +Firstly, they are used for the definition of the motion of the Mobile platform with respect to the fixed frame: +
+-
+
- In position: \({}^A\bm{P}_{B}\) (read: Position of frame \(\{B\}\) expressed in frame \(\{A\}\)) +
- In rotation: \({}^A\bm{R}_{B}\) (read: The rotation matrix that express the orientation of frame \(\{B\}\) expressed in frame \(\{A\}\)) +
+The frames \(\{A\}\) and \(\{B\}\) are used for all the kinematic analysis (Jacobian, Stiffness matrix, …). +
+ ++Typical choice of \(\{A\}\) and \(\{B\}\) are: +
+-
+
- Center of mass of the payload +
- Location where external forces are applied to the mobile platform (for instance when the mobile platform is in contact with a stiff environment) +
- Center of the cube for the cubic configuration +
+The definition of the frames is done with the initializeFramesPositions function (link);
+
+
Figure 1: Definition of the Frames for the Stewart Platform
+1.2 Location of the Spherical Joints
++Then, we define the location of the spherical joints (see Figure 2):
- \(\bm{a}_{i}\) are the position of the spherical joints fixed to the fixed base @@ -427,49 +497,180 @@ Then, we define the location of the spherical joints:
-We define the rest position of the Stewart platform: +The location of the joints will define the Geometry of the Stewart platform. +Many characteristics of the platform depend on the location of the joints. +
+ +
+The location of the joints can be set to arbitrary positions (function generateGeneralConfiguration described here) or can be computed to obtain specific configurations such as:
-
-
- For simplicity, we suppose that the fixed base and the moving platform are parallel and aligned with the vertical axis at their rest position. -
- Thus, to define the rest position of the Stewart platform, we just have to defined its total height \(H\). -\(H\) corresponds to the distance from the bottom of the fixed base to the top of the moving platform. +
- A cubic configuration: function
generateCubicConfiguration(link).
+ - A symmetrical configuration
-From \(\bm{a}_{i}\) and \(\bm{b}_{i}\), we can determine the length and orientation of each strut: +The location of the spherical joints are then given by \({}^{F}\bm{a}_{i}\) and \({}^{M}\bm{b}_{i}\). +
+ + +
+
Figure 2: Position of the Spherical/Universal joints for the Stewart Platform
+1.3 Length and orientation of the struts
++From the location of the joints (\({}^{F}\bm{a}_{i}\) and \({}^{M}\bm{b}_{i}\)), we compute the length \(l_i\) and orientation of each strut \(\hat{\bm{s}}_i\) (unit vector aligned with the strut). +The length and orientation of each strut is represented in figure 3. +
+ +
+This is done with the computeJointsPose function (link).
+
+
Figure 3: Length \(l_i\) and orientation \(\hat{\bm{s}}_i\) of the Stewart platform struts
+1.4 Rest Position of the Stewart platform
++We may want to initialize the Stewart platform in some position and orientation that corresponds to its rest position. +
+ ++To do so, we choose:
-
-
- \(l_{i}\) is the length of the strut -
- \({}^{A}\hat{\bm{s}}_{i}\) is the unit vector align with the strut +
- the position of \(O_B\) expressed in \(\{A\}\) using \({}^AP\) +
- the orientation of \(\{B\}\) expressed in \(\{A\}\) using a rotation matrix \({}^{A}R_{B}\)
-The position of the Spherical joints can be computed using various methods:
+Then, the function initializeStewartPose (link) compute the corresponding initial and rest position of each of the strut.
+
+
