Force-position hybrid control method in decoupling joint space of eight-cable parallel mechanism

Abstract

The invention provides a force-position hybrid control method in a decoupling joint space of an eight-cable parallel mechanism, which comprises the following steps: in the whole eight-cable parallel mechanism, a frame is mounted on a base, a driving unit comprises a motor, a reel, a tension sensor and a pulley, the reel is connected with the motor, the tension sensor, the reel and the motor are locked on a mounting plate, and the pulley is connected with the motor. One end of any rope is connected with one vertex of the motion platform through a pulley, and the other end of the rope is connected with the reel through a tension sensor; according to the force-position hybrid control method in the decoupling joint space of the eight-rope parallel mechanism, position control of a motion platform and contact force control of the environment are decoupled through a mechanism kinematics model and statics equilibrium conditions, and rope length control and rope tension control are conducted on ropes respectively. According to the force-position hybrid control method in the decoupling joint space of the eight-cable parallel mechanism, position control and force control are conducted on the cable parallel mechanism at the same time, and various contact operation requirements are met.

Description

Technical Field

[0001] This invention belongs to the field of compliant control technology for parallel robots, and particularly relates to a force-position hybrid control method for decoupled joint space of an eight-cable parallel mechanism. Background Technology

[0002] Force-position hybrid control of cable parallel mechanisms, ensuring they maintain desired contact force with the environment while moving along it, is a key technology in the compliance control of cable parallel mechanisms to meet the requirements of contact operations such as grinding, polishing, and assembly. Cable parallel mechanisms have advantages such as simple structure, light weight, large working space, and high load-bearing capacity, and are widely used in an increasing number of fields, such as lifting, aircraft wind tunnel testing, and human rehabilitation. When using cable parallel mechanisms for contact operations such as grinding, polishing, and assembly, force-position hybrid control is required to ensure they maintain desired contact force with the environment while moving along it, thereby meeting the operational requirements.

[0003] In existing force-position hybrid control methods for cable parallel mechanisms, the contact force control between the motion platform and the environment is achieved based on admittance control. When the environmental stiffness is high and the motion platform has position control errors, it is difficult to accurately control the contact force between the motion platform and the environment. Excessive contact force can easily damage the mechanism itself or the external environment, while insufficient contact force can easily lead to failures in grinding, polishing, assembly, and other operations. Therefore, how to achieve effective and reliable force-position hybrid control of cable parallel mechanisms, enabling them to meet position control requirements while achieving precise contact force control, is a pressing technical problem that needs to be solved. Summary of the Invention

[0004] In view of this, the present invention aims to propose a force-position hybrid control method in the decoupled joint space of an eight-cable parallel mechanism, in order to solve the problem of how to effectively and reliably control the force-position hybrid control of the cable parallel mechanism, so as to achieve precise contact force control while meeting the position control requirements.

[0005] To achieve the above objectives, the technical solution of the present invention is implemented as follows:

[0006] This invention provides a force-position hybrid control method for an eight-rope parallel mechanism decoupled within the joint space. The eight-rope parallel mechanism comprises a drive unit, ropes, a frame, a motion platform, a base, and a mounting plate. The frame is mounted on the base. The drive unit includes a motor, a reel, a tension sensor, and a pulley. The reel is connected to the motor. The tension sensor, the reel, and the motor are locked to the mounting plate. One end of any rope is connected to a vertex of the motion platform via a pulley, and the other end is connected to the reel via a tension sensor. The force-position hybrid control method for the eight-rope parallel mechanism decouples the position control and the contact force control with the environment of the motion platform through a kinematic model of the mechanism and static equilibrium conditions, and separately controls the rope length and rope tension.

[0007] Furthermore, the specific process of the force-position hybrid control method for the decoupled joint space of the eight-cable parallel mechanism includes step S1, establishing a global coordinate system {O, X, Y, Z} and a local coordinate system {P, x, y, z}, wherein the local coordinate system {P, x, y, z} is a coordinate system established at the center of mass point P of the motion platform, B i P is the point of origin. i This is the connection point between the rope and the sports platform. O b i Let O be the position vector of the rope exit point in the global coordinate system {O, X, Y, Z}. O p is the position vector of the origin of the local coordinate system in the global coordinate system {O, X, Y, Z}. i Let l be the position vector of the connection point between the rope and the motion platform in the local coordinate system {P, x, y, z}. i Let P be the rope vector. i B i , O R P Let l be the rotation matrix from the local coordinate system to the global coordinate system, then we have, l i = O b i - O p- O R p p i (1); In equation (1), i is the rope number, i = 1, 2, ..., 8; the unit direction vector of the rope is: In equation (2), i is the rope number, i = 1, 2, ..., 8; then, according to the inverse kinematics of the cable parallel mechanism, the length of each rope can be obtained by solving equation (1). When the motion platform comes into contact with the environment, the motor controls the position and posture of the motion platform by controlling the length of the five ropes and with the other three ropes being tensioned.

[0008] Furthermore, while controlling the length of 5 ropes, the remaining 3 ropes are tension-controlled to achieve the purpose of controlling the contact force between the motion platform and the environment. The specific control process is as follows: Step S1, establish the mechanical model of the eight-rope parallel mechanism and establish the static equilibrium formula based on the force situation of the motion platform; Step S2, based on the force situation of the eight-rope parallel mechanism when it moves along the contact environment, establish the force balance equation of the eight-rope parallel mechanism, and generate a function related to the contact force with the contact environment by combining the static equilibrium formula; Step S3, set the desired contact force between the motion platform and the environment, solve the force balance equation, obtain the desired tension of each rope, and control the tension of the remaining 3 ropes other than the 5 rope length control ropes.

[0009] Furthermore, step S1 specifically involves the following steps: Let t i =t i u i , t i Let be the rope tension value, in N, where i is the rope number, i = 1, 2, ..., 8; the established static equilibrium formula is: (4); where, in equation (3), f p The external force acting on the motion platform is expressed in N; in equation (4), τ p The external torque acting on the motion platform is expressed in N·m; combining equations (3) and (4), we obtain J. T T+F=0(5); In equation (5), J T For structure matrix, Let T be a 6x8 matrix of real numbers; T is the rope tension matrix, T = [t1...t8]. T , Represents an 8x1 matrix of real numbers relating to the rope tension; F is the spinor of the forces acting on the motion platform, F = [f p τ p ] T .

[0010] Furthermore, step S2 specifically involves the following steps: J T T+G+f c +f f +τ f =0(6); In equation (6), τ f Friction force f f The torque on the motion platform, expressed in N·m; f c ρ is the contact force between the motion platform and the contact environment, in N; G is the gravity acting on the motion platform, in N; where f f τ f f f The relationship between them is: In equation (7), r is the vertical distance from the center of mass of the moving platform to the contact environment, and the magnitude of the vector μ is equal to the friction coefficient μ between the moving platform and the environment, and its direction is opposite to the direction of motion of the moving platform; substituting equation (7) into equation (6), we get:

[0011] J T T+G+f c +μf c +r×μf c =0(8); Moving the second term and all subsequent terms in equation (8) to the right side of the equation, all terms on the right side can be expressed as contact force f. c The function, i.e., F = (f c ): J T T = -(G + f c +μf c +r×μf c )=F(f c (9); Expanding equation (9) yields equation (10): (10); Equation (10) contains 9 unknowns, namely t1~t8, f c Set f c Given the desired contact value, and by solving equation (10), the desired tension of each rope is obtained. At this point, with the tension of 3 ropes controlled and the length of the remaining 5 ropes determined, the tension of all ropes and the contact force between the motion platform and the environment can be determined.

[0012] Furthermore, for a parallel cable mechanism with n degrees of freedom driven by m cables, equation (9) expands to n unknowns containing m+1 unknowns (t1, ..., t). m f c The equations require the motor encoder on the motor to control the length of n-1 ropes and the tension of the remaining m-n+1 ropes. The tension sensor is used to provide real-time feedback on the actual tension of the ropes, which can achieve the purpose of simultaneously controlling the position of the m-n degree-of-freedom parallel mechanism and the contact force with the environment.

[0013] Compared with the prior art, the force-position hybrid control method in the decoupled joint space of the eight-cable parallel mechanism described in this invention has the following advantages:

[0014] (1) The force-position hybrid control method in the decoupled joint space proposed in this invention can simultaneously control the position and force of the cable parallel mechanism, which can meet the needs of contact operations such as grinding, polishing, and assembly.

[0015] (2) Because the control of the rope tension can achieve high precision, the control of the contact force between the motion platform and the environment can also be highly precise. This force control effect in force-position hybrid control by controlling the rope tension is more accurate than the force control effect of the existing force-position hybrid control method based on admittance control.

[0016] (3) The force-position hybrid control method in the decoupled joint space proposed in this invention can be applied to various types of cable-driven parallel mechanisms. For an n-degree-of-freedom cable parallel mechanism driven by m cables, by controlling the length of n-1 cables and controlling the tension of the remaining m-n+1 cables, the position of the motion platform of the cable parallel mechanism and its contact force with the environment can be controlled simultaneously. Attached Figure Description

[0017] The accompanying drawings, which form part of this invention, are used to provide a further understanding of the invention. The illustrative embodiments of the invention and their descriptions are used to explain the invention and do not constitute an improper limitation of the invention.

[0018] In the attached diagram:

[0019] Figure 1 This is an isometric schematic diagram of the eight-cable parallel mechanism described in an embodiment of the present invention;

[0020] Figure 2 This is an isometric view of the drive unit in the eight-wire parallel mechanism described in an embodiment of the present invention;

[0021] Figure 3 This is a simplified schematic diagram of the eight-wire parallel mechanism in the force-position hybrid control method for decoupling the joint space of the eight-wire parallel mechanism described in the embodiment of the present invention;

[0022] Figure 4 This is a schematic diagram illustrating the coordinates of the connection points between the ropes and the motion platform in the force-position hybrid control method within the decoupled joint space of the eight-rope parallel mechanism described in this embodiment of the invention.

[0023] Figure 5 This is a schematic diagram of the kinematic model of the eight-wire parallel mechanism in the schematic diagram of the force-position hybrid control method in the decoupled joint space of the eight-wire parallel mechanism described in the embodiment of the present invention;

[0024] Figure 6 This is a schematic diagram of the mechanical model of the eight-wire parallel mechanism in the schematic diagram of the force-position hybrid control method in the decoupled joint space of the eight-wire parallel mechanism described in the embodiment of the present invention;

[0025] Figure 7 This is a schematic diagram of the force analysis of the eight-wire parallel mechanism in the schematic diagram of the force-position hybrid control method in the decoupling joint space of the eight-wire parallel mechanism according to an embodiment of the present invention;

[0026] Figure 8This is a schematic diagram of the control block diagram for the force-position hybrid control method in the decoupled joint space of the cable parallel mechanism according to an embodiment of the present invention.

[0027] Explanation of reference numerals in the attached figures:

[0028] 1. Drive unit; 2. Rope; 3. Frame; 4. Motion platform; 5. Base; 6. Motor; 7. Mounting plate; 8. Pulley; 9. Tension sensor; 10. Reel. Detailed Implementation

[0029] It should be noted that, unless otherwise specified, the embodiments and features described in the present invention can be combined with each other.

[0030] In the description of this invention, it should be understood that the terms "center," "longitudinal," "lateral," "upper," "lower," "front," "rear," "left," "right," "vertical," "horizontal," "top," "bottom," "inner," and "outer," etc., indicating orientations or positional relationships based on the orientations or positional relationships shown in the accompanying drawings, are only for the convenience of describing the invention and simplifying the description, and do not indicate or imply that the device or element referred to must have a specific orientation, or be constructed and operated in a specific orientation, and therefore should not be construed as a limitation of the invention. Furthermore, the terms "first," "second," etc., are used for descriptive purposes only and should not be construed as indicating or implying relative importance or implicitly specifying the number of indicated technical features. Thus, a feature defined with "first," "second," etc., may explicitly or implicitly include one or more of that feature. In the description of this invention, unless otherwise stated, "a plurality of" means two or more.

[0031] In the description of this invention, it should be noted that, unless otherwise explicitly specified and limited, the terms "installation," "connection," and "linking" should be interpreted broadly. For example, they can refer to a fixed connection, a detachable connection, or an integral connection; they can refer to a mechanical connection or an electrical connection; they can refer to a direct connection or an indirect connection through an intermediate medium; and they can refer to the internal connection of two components. Those skilled in the art will understand the specific meaning of the above terms in this invention based on the specific circumstances.

[0032] The present invention will now be described in detail with reference to the accompanying drawings and embodiments.

[0033] See Figures 1-7As shown, this embodiment provides a force-position hybrid control method for decoupling the joint space of an eight-rope parallel mechanism. The eight-rope parallel mechanism includes a drive unit 1, ropes 2, a frame 3, a motion platform 4, a base 5, and a mounting plate 7. The frame 3 is mounted on the base 5. The drive unit 1 includes a motor 6, a reel 10, a tension sensor 9, and a pulley 8. The reel 10 is connected to the motor 6. The tension sensor 9, the reel 10, and the motor 8 are locked to the mounting plate 5. One end of any rope 2 is connected to a vertex of the motion platform 4 via the pulley 8, and the other end is connected to the reel 10 via the tension sensor 9. The force-position hybrid control method for decoupling the joint space of the eight-rope parallel mechanism decouples the position control of the motion platform and the contact force control with the environment through the kinematic model of the mechanism and static equilibrium conditions. The rope length and rope tension are controlled separately, thereby achieving the purpose of simultaneously controlling the position of the motion platform and its contact force with the environment.

[0034] Specifically, in this embodiment, the specific process of the force-position hybrid control method in the decoupled joint space of the eight-cable parallel mechanism includes step S1, establishing a global coordinate system {O, X, Y, Z} and a local coordinate system {P, x, y, z}, wherein the local coordinate system {P, x, y, z} is a coordinate system established at the center of mass point P of the motion platform, B i P is the point of origin. i This is the connection point between the rope and the sports platform. O b i Let be the position vector of the rope exit point in the global coordinate system {0, X, Y, Z}. O p is the position vector of the origin of the local coordinate system in the global coordinate system {O, X, Y, Z}. i Let l be the position vector of the connection point between the rope and the motion platform in the local coordinate system {P, x, y, z}. i Let P be the rope vector. i B i , O R P Let l be the rotation matrix from the local coordinate system to the global coordinate system, then we have, l i = O b i - O p- O R p p i (1); In equation (1), i is the rope number, i = 1, 2, ..., 8; the unit direction vector of the rope is: In equation (2), i is the rope number, i = 1, 2, ..., 8; then, according to the inverse kinematics of the cable parallel mechanism, the length of each rope can be obtained by solving equation (1). When the motion platform comes into contact with the environment, the motor can control the position and posture of the motion platform by controlling the length of the five ropes (when the other three ropes are all tensioned).

[0035] Specifically, in this embodiment, with the length of 5 ropes controlled (i.e., the position of the motion platform is determined), tension control is applied to the remaining 3 ropes to achieve the purpose of controlling the contact force between the motion platform and the environment. The specific control process is as follows: Step S1, establish a mechanical model of the eight-rope parallel mechanism and establish a static equilibrium formula based on the force conditions of the motion platform; Step S2, establish a force balance equation for the eight-rope parallel mechanism based on the force conditions of the eight-rope parallel mechanism when it moves along the contact environment, and generate a function related to the contact force with the contact environment by combining the static equilibrium formula; Step S3, set the desired contact force between the motion platform and the environment, solve the force balance equation, obtain the desired tension of each rope, and apply tension control to the remaining 3 ropes other than the 5 rope length control ropes.

[0036] Specifically, in this embodiment, step S1 is as follows: Let t i =t i u i , t i Let be the rope tension value, in N, where i is the rope number, i = 1, 2, ..., 8; the established static equilibrium formula is: In equation (3), f p The external force acting on the motion platform is expressed in N; in equation (4), τ p The external torque acting on the motion platform is expressed in N·m; combining equations (3) and (4), we obtain J. T T+F=0(5); In equation (5), J T For structure matrix, Let T be a 6x8 matrix of real numbers; T is the rope tension matrix, T = [t1...t8]. T , Represents an 8x1 matrix of real numbers relating to the rope tension; F is the spinor of the forces acting on the motion platform, F = [f p τ p ] T .

[0037] Specifically, in this embodiment, step S2 consists of the following steps: J T T+G+f c +f f +τ f =0(6); In equation (6), τ f Friction force f f The torque on the motion platform, expressed in N·m; f c ρ is the contact force between the motion platform and the contact environment, in N; G is the gravity acting on the motion platform, in N; where f f τ f ff The relationship between them is: (7); In equation (7), r is the vertical distance from the center of mass of the moving platform to the contact environment, and the magnitude of the vector μ is equal to the friction coefficient μ between the moving platform and the environment, and its direction is opposite to the direction of motion of the moving platform; Substituting equation (7) into equation (6), we get: J T T+G+f c +μf c +r×μf c =0(8); Moving the second term and all subsequent terms in equation (8) to the right side of the equation, all terms on the right side can be expressed as contact force f. c The function, i.e., F = (f c ): J T T = -(G + f c +μf c +r×μf c )=F(f c (9); Expanding equation (9) yields equation (10): Equation (10) contains 9 unknowns, namely t1~t8, f c Set f c The desired contact value is obtained by solving equation (10) to obtain the desired tension of each rope. At this time, if the tension of 3 ropes is controlled and the length of the remaining 5 ropes is determined, the tension of all ropes and the contact force between the motion platform and the environment can be determined.

[0038] Specifically, in this embodiment, for the n-degree-of-freedom cable parallel mechanism driven by m ropes, equation (9) is expanded into a set of n equations containing m+1 unknowns. The length of n-1 ropes needs to be controlled by the motor encoder on the motor, the tension of the remaining m-n+1 ropes needs to be controlled, and the actual tension of the ropes needs to be fed back in real time by the tension sensor.

[0039] Position control of the motion platform is achieved by controlling the rope lengths. Given the desired trajectory of the motion platform, the desired lengths and velocities of each rope can be obtained through real-time inverse kinematics solving of the parallel cable mechanism. A rope length controller then controls the rotation of n-1 servo motors, which in turn drive the reels to pull the corresponding ropes, producing the appropriate motion. When the motion platform comes into contact with the environment, pose control of the platform's n degrees of freedom can be achieved by controlling the lengths of n-1 of the m ropes. Motor encoders provide real-time feedback of the actual rope lengths, completing the rope length control closed loop. It is worth noting that due to elastic deformation of the ropes, measuring the actual rope length using a motor encoder will introduce some error. To improve the accuracy of rope length feedback, optical measuring elements can be used to measure the pose information of the motion platform, and the precise lengths of each rope can be obtained through inverse kinematics solving.

[0040] With the position of the motion platform determined and the desired contact force between the platform and the environment known, the desired tension of each rope can be calculated (the research on methods for solving rope tension in parallel cable mechanisms is quite mature and will not be elaborated here). The remaining m-n+1 servo motors are controlled by a rope tension controller to generate torque, thereby driving the reel to pull the corresponding ropes to produce the desired tension. Tension sensors are used to provide real-time feedback on the actual rope tension, thus completing the closed-loop rope tension control. As shown above, with the lengths of n-1 ropes determined, if the tension of the remaining m-n+1 ropes is controlled, the tension of all ropes and the contact force between the motion platform and the environment can be determined.

[0041] The above two steps are performed simultaneously, thereby achieving the purpose of simultaneously controlling the position of the motion platform and its contact force with the environment. The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A method for force-position hybrid control within the decoupled joint space of an eight-cable parallel mechanism, characterized in that, The eight-rope parallel mechanism comprises a drive unit, ropes, a frame, a motion platform, a base, and a mounting plate. The frame is mounted on the base. The drive unit includes a motor, a reel, a tension sensor, and a pulley. The reel is connected to the motor. The tension sensor, the reel, and the motor are locked to the mounting plate. One end of each rope is connected to a vertex of the motion platform via a pulley, and the other end is connected to the reel via a tension sensor. The force-position hybrid control method of the eight-rope parallel mechanism decouples the position control of the motion platform from the contact force control with the environment through the kinematic model of the mechanism and static equilibrium conditions, and separately controls the rope length and rope tension.

2. The force-position hybrid control method in the decoupled joint space of the eight-cable parallel mechanism according to claim 1, characterized in that, The specific process of the force-position hybrid control method for decoupling the joint space of the eight-cable parallel mechanism includes step S1, establishing a global coordinate system {O, X, Y, Z} and a local coordinate system {P, x, y, z}, wherein the local coordinate system {P, x, y, z} is a coordinate system established at the center of mass point P of the motion platform, B i P is the point of origin. i This is the connection point between the rope and the sports platform. O b i Let O be the position vector of the rope exit point in the global coordinate system {O, X, Y, Z}. O p is the position vector of the origin of the local coordinate system in the global coordinate system {O, X, Y, Z}. i Let l be the position vector of the connection point between the rope and the motion platform in the local coordinate system {P, x, y, z}. i Let P be the rope vector. i B i , O R P Let l be the rotation matrix from the local coordinate system to the global coordinate system, then we have, i = O b i - O p- O R p p i (1); In equation (1), i is the rope number, i = 1, 2, ..., 8; the unit direction vector of the rope is: In equation (2), i is the rope number, i = 1, 2, ..., 8; then, according to the inverse kinematics of the cable parallel mechanism, the length of each rope can be obtained by solving equation (1). When the motion platform comes into contact with the environment, the motor controls the position and posture of the motion platform by controlling the length of the five ropes and with the other three ropes being tensioned.

3. The force-position hybrid control method in the decoupled joint space of the eight-cable parallel mechanism according to claim 2, characterized in that, With length control of 5 ropes, tension control of the remaining 3 ropes is implemented to control the contact force between the motion platform and the environment. The specific control process is as follows: Step S1, establish the mechanical model of the eight-rope parallel mechanism and establish the static equilibrium formula based on the force conditions of the motion platform; Step S2, establish the force balance equation of the eight-rope parallel mechanism based on the force conditions of the eight-rope parallel mechanism when it moves along the contact environment, and generate a function related to the contact force with the contact environment by combining the static equilibrium formula; Step S3, set the desired contact force between the motion platform and the environment, solve the force balance equation, obtain the desired tension of each rope, and apply tension control to the remaining 3 ropes other than the 5 rope length control ropes.

4. The force-position hybrid control method in the decoupled joint space of the eight-cable parallel mechanism according to claim 2, characterized in that, Step S1 is as follows: Let t i =t i u i , t i Let be the rope tension value, in N, where i is the rope number, i = 1, 2, ..., 8; the established static equilibrium formula is: In equation (3), f p The external force acting on the motion platform is expressed in N; in equation (4), τ p The external torque acting on the motion platform is expressed in N·m; combining equations (3) and (4), we obtain J. T T+F=0(5); In equation (5), J T For structure matrix, J T ∈R1 6×8 R1 6×8 Let T be a 6x8 real matrix; T is the rope tension matrix, T = [t1...t8]. T , T∈R2 8×1 , Represents an 8x1 matrix of real numbers relating to the rope tension; F is the spinor of the forces acting on the motion platform, F = [f p τ p ] T .

5. The force-position hybrid control method in the decoupled joint space of the eight-cable parallel mechanism according to claim 4, characterized in that, Step S2 is as follows: J T T+G+f c +f f +τ f =0(6); In equation (6), τ f Friction force f f The torque on the motion platform, expressed in N·m; f c ρ is the contact force between the motion platform and the contact environment, in N; G is the gravity acting on the motion platform, in N; where f f τ f f f The relationship between them is: In equation (7), r is the vertical distance from the center of mass of the moving platform to the contact environment, and the magnitude of the vector μ is equal to the friction coefficient μ between the moving platform and the environment, and its direction is opposite to the direction of motion of the moving platform; substituting equation (7) into equation (6), we get: J T T+G+f c +μf c +r×μf c =0(8); Moving the second term and all subsequent terms in equation (8) to the right side of the equation, all terms on the right side can be expressed as contact force f. c The function, i.e., F = (f c ): J T T = -(G + f c +μf c +r×μf c )=F(f c (9); Expanding equation (9) yields equation (10): Equation (10) contains 9 unknowns, namely t1~t8, f c Set f c The desired contact value is obtained by solving equation (10) to obtain the desired tension of each rope. At this time, if the tension of 3 ropes is controlled and the length of the remaining 5 ropes is determined, the tension of all ropes and the contact force between the motion platform and the environment can be determined, thereby achieving the purpose of controlling the contact force between the motion platform and the environment.

6. The force-position hybrid control method in the decoupled joint space of the eight-cable parallel mechanism according to claim 5, characterized in that, For a parallel cable mechanism with n degrees of freedom driven by m cables, equation (9) expands to n unknowns containing m+1 unknowns (t1, ..., t). m f c The equations require controlling the length of n-1 ropes and the tension of the remaining m-n+1 ropes through the motor encoder on the motor, and using a tension sensor to provide real-time feedback on the actual tension of the ropes.

Images