Method and apparatus for controlling geometrically simple parallel mechanisms with distinctive connections

A motion system capable of motion in six degrees of freedom, comprising:

• a base platform comprising at least three points defining a base plane, said three points in said base plane defining a first triangle;
• a movable platform providing at least three points defining a second plane, said three points in said second plane defining a second triangle;
• at least six linearly extensible support members extending between said base platform and said movable platform,
• each of said support members being actuatable for a motion-producing system and/or alternatively being passively-force resistant for a motion-sensing system,
• each of said support members being connected (1) from a first single point of connection on a line defined by any two of said at least three points in said first plane on said base platform, said first single point of connection being distinct from any other first single point of connection of any other support member, (2) to a corresponding second single point of connection on a line defined by any two of said at least three points in said second plane on said second triangle, said second single point of connection being distinct from any other second single point of connection of any other support member, for imparting or allowing motion of said movable platform relative to said base platform in accordance with selective changes in the lengths of said support members,
• at least three of said support members being connected at the apices of said first triangle and at least three of said support members being connected at the apices of said second triangle;
• and
• control means, coupled to said linearly extensible support members, for actuating or responsive to selective changes in the lengths of said support members, and for producing predictable movement of said movable platform when said support members are actuatable or sensing movement of said movable platform when said support members are passively force-resistant.

An apparatus for controlling and/or determining the position and orientation of a movable platform relative to a base platform, having linear extensible members extending between said movable platform and said base platform, given predetermined dimensions for said movable platform, said base platform, and lengths of lines of a structure formed by said movable platform, said base platform, and said linearly extensible support members, said structure being reducible to an equivalent octahedron, said lines comprising virtual octahedron edges extending between said movable platform and said base platform, the apparatus comprising:

• transducer means for acquiring the lengths of said lines;
• computing means responsive to said acquired line lengths for computing at least one closed-form forward displacement solution of said structure;
• and
• determining means responsive to said at least one closed-form forward displacement solution for determining the position and orientation of said movable platform relative to said base platform.

A method for controlling and/or determining, with a computer, the position and orientation of a movable platform relative to a base platform, having linear extensible members extending between said movable platform and said base platform, given predetermined dimensions for said movable platform, said base platform, and lengths of lines of a structure formed by said movable platform, said base platform, and said linearly extensible support members, said structure being reducible to an equivalent octahedron, said lines comprising virtual octahedron edges extending between said movable platform and said base platform, the method comprising the steps of:

• (1) acquiring the lengths of said lines;
• (2) utilizing said acquired line lengths, computing at least one closed-form forward displacement solution of the geometry of said structure;
• and
• (3) utilizing said at least one closed-form forward displacement solution, determining the position and orientation of said movable platform relative to said base platform.

A method for controlling and determining, with a computer, the position and orientation of a Stewart-type platform, given predetermined dimensions for a movable platform having a predetermined number of apices, a base platform having a predetermined number of apices, and leg lengths of variable positionable linear actuating means extending between said movable platform and said base platform, said leg lengths being determined by corresponding linear position transducers, comprising the steps of:

• (1) acquiring from said linear position transducers the lengths of said variably positionable linear actuating means to obtain known leg lengths;
• (2) storing in a memory data corresponding to a path of travel for said movable platform;
• (3) utilizing said known leg lengths, determining the constants corresponding to spherical four-bar linkages associated with each apex of said base platform;
• (4) utilizing said constants, determining a four-bar polynomial associated with each of said spherical four-bar linkages of said base platform;
• (5) determining an eliminant polynomial corresponding to each of said four-bar polynomials;
• (6) calculating all real roots of said eliminant polynomial;
• (7) identifying one of said real roots of said eliminant polynomial based on a comparison of said real roots to said stored path of travel data;
• and
• (8) providing to a controller said identified one real root of said eliminant as one angular coordinate of said movable platform relative to said base platform, said controller being adapted to use said one real root to generate positional feedback information and to control the position of said movable platform in response to said positional feedback information.

A method for controlling and determining, with a computer, the position and orientation of a Stewart-type equivalent parallel mechanism, given predetermined dimensions for a movable platform, a base platform, and leg lengths of variably positionable linear actuating members extending between said movable platform and said base platform, said leg lengths being determined by corresponding linear position transducers, comprising the steps of:

• (1) acquiring from said linear position transducers the lengths of said variably positionable linear actuating members to obtain known leg lengths;
• (2) storing in memory data corresponding to a path of travel for said movable platform;
• (3) assembling twelve edge lengths of a virtual octahedron corresponding to said movable platform, said base platform, and said known leg lengths;
• (4) utilizing said twelve edge lengths, determining spherical four-bar linkages associated with the apices of said base platform;
• (5) computing a set of constant coefficients of an input-output relation corresponding to each of said spherical four-bar linkages associated with said base platform apices;
• (6) assembling three polynomial equations in three variables representing angular input/output relationships for said spherical four-bar linkages, each of said three polynomial equations being bi-quadric and containing two of said three variables,
• (7) obtaining an eliminant polynomial in one of said three variables whose roots comprise all possible values of said one variable which exist in all possible solution sets of said three variables for said three bi-quadric polynomials;
• (8) calculating all real roots of said eliminant polynomial;
• (9) identifying one of said real roots of said eliminant polynomial based on predetermined root selection criteria;
• (10) providing said identified one real root of said eliminant as one angular coordinate of said movable platform relative to said base platform;
• (11) determining the remaining angular coordinates of said movable platform utilizing said first calculated coordinate;
• (12) determining Cartesian coordinates of said movable platform relative to said base platform utilizing said angular coordinates;
• and
• (13) providing said Cartesian coordinates of said movable platform as positional feedback to a controller adapted to control the position of said movable platform in response to said positional feedback.

A system for controlling and determining the position and orientation of a Stewart-type platform, given predetermined dimensions for a movable platform, a base platform, and leg lengths of variably positionable linear actuating members extending between said movable platform and said base platform, comprising:

• transducer means for acquiring the lengths of said variably positionable linear actuating members to obtain known leg lengths;
• means for storing in memory data corresponding to a path of travel for said movable platform;
• means for assembling data corresponding to the edge lengths of an octahedron formed by said movable platform, said base platform, and said known leg lengths;
• means responsive to said edge length data for determining spherical four-bar linkages associated with the apices of said base platform;
• means for computing a set of constant coefficients of an input-output relation corresponding to each of said spherical four-bar linkages associated with said base platform apices;
• means for assembling three polynomial equations in three variables representing angular input/output relationships for said spherical four-bar linkages, each of said three polynomial equations being bi-quadric and containing two of said three variables,
• means for obtaining an eliminant polynomial in one of said three variables whose roots comprise all possible values of said one variables which exist in all possible solution sets of said three variables for said three bi-quadric polynomials;
• means for calculating all real roots of said eliminant polynomial;
• means for identifying one of said real roots of said eliminant polynomial based on predetermined root selection criteria;
• means for providing said identified one real root of said eliminant as one angular coordinate of said movable platform relative to said base platform;
• means responsive to said one angular coordinate for determining the remaining angular coordinates of said movable platform;
• means responsive to said angular coordinates for determining Cartesian coordinates of said movable platform relative to said base platform;
• and
• feedback means for providing said Cartesian coordinates of said movable platform as positional feedback to a controller adapted to control the position of said movable platform in response to said positional feedback.

A method for controlling and/or determining the position and orientation of a movable platform relative to a base platform, with each one of six linear extensible members connected between a distinct point of connection on said base platform and a distinct point of connection on said movable platform, given predetermined dimensions for said movable platform and said base platform, the method comprising the steps of:

• (1) determining an equivalent octahedron structure based on a structure comprising said linear extensible members, said movable platform, and said base platform, the equivalent octahedron structure including virtual lines extending between said movable platform and said base platform;
• (2) acquiring the lengths of said virtual lines;
• (3) utilizing said acquired line lengths, computing at least one closed-form forward displacement solution of the geometry of said equivalent geometric structure;
• and
• (4) utilizing said at least one closed-form forward displacement solution, determining the position and orientation of said movable platform relative to said base platform.

A control system for use in a closed-loop Cartesian controller for a parallel mechanism having a movable platform, a base platform, and a plurality of linearly extensible legs supporting said movable platform relative to said base platform, comprising:

• means for receiving Cartesian coordinate position commands indicative of a commanded Cartesian coordinate position and orientation for said movable platform from a higher level control source external to said control system;
• reverse displacement control means responsive to said Cartesian coordinate position commands for providing leg length commands corresponding to a desired length for said legs;
• leg control means responsive to said leg length commands for actuating selective changes in the lengths of said legs;
• means for sensing the lengths of said legs and for providing actual leg length signals;
• and
• closed-form forward displacement control means responsive to said actual leg length signals for determining and providing Cartesian coordinate actual position and orientation signals of said movable platform as feedback to said higher level control source,
• thereby forming a closed-loop Cartesian controller for said movable platform.