1. 研究目的与意义（文献综述包含参考文献）

Trajectory planning of anthropomorphic arm Abstract In this paper, an algorithm for the direction control of endpoint position error caused by disturbance is proposed. With this in mind the effect of the disturbance on the position of the endpoint is maximized in the direction of the singular vector corresponding to the maximum singular value. Therefore, if we can control the direction of the singular vector, we can control the direction of the endpoint position error guided by the disturbance. The algorithm is applied to the hit motion of the manipulator, and the control algorithm and effect of the position error direction are presented. Trajectory planning of anthropomorphic arm Introduction In general, we cannot get the nominal model of real robot system, because real robot system contains various uncertainties. One uncertainty is noise and interference, which are excluded from the nominal model. For example, sensor noise and angular velocity quantization errors act as disturbances on the robot. In order to suppress disturbances, many scholars focus their research on the design of robust controllers [2] and [7]. The design of the controller is very complicated. At the same time, our concept is to effectively use the dynamics of the robot to complete the task of robustness, even if the controller is very simple. Therefore, we focus on motion planning rather than controller design. If the robust trajectory can be obtained independently of the control input, the controller can be simplified. In order to reduce the influence of disturbance on the task, we use the dynamic characteristics of the robot to plan the motion proactively. As far as we know, although the motion planning of [1], [9], [8] and other mechanical arms has been proposed, these schemes do not involve the problem of disturbance suppression. The results show that this method can effectively reduce the position error of the final point. On the other hand, this paper focuses on the direction of endpoint position error. In some tasks, the direction in which the position error is generated is more important than its size. One of these tasks is the strike motion of the robotic arm, as shown in figure 1. If the endpoint is disturbed and deviates vertically to the desired trajectory, the robot cannot hit the target, as shown in figure 1(a). If the direction of the position error can be controlled to be tangent to the desired trajectory, as shown in the figure. 1(b),Even with interference, the manipulator can hit the target. For hitting motion, there is no need to force the manipulator to reduce the size of position error. As long as the direction of position error can be controlled, the robustness of batting motion against disturbance can be improved. Therefore, this paper only deals with the generating direction of endpoint position error, and proposes a technology to control the generating direction to the tangential direction of the target path. Fig. 1. Application examples of the control of direction of the end-point position error caused by disturbance Related works In the traditional motion planning of industrial robots and redundant robots, the task is usually simple and fixed, and the planning is realized by optimizing predetermined performance indexes such as dexterity, joint limitation, torque limitation and obstacle avoidance With the gradual maturity of industrial manipulator technology, production efficiency is greatly improved, and the demand for robots in factories is also gradually reduced. Therefore, the research focus in the field of robotics has shifted from industrial robots [7] to service robots, and humanoid robots play a very important role in the field of service robots. Due to the various requirements faced by humanoid robots, the humanoid arm must be able to handle more complex task-level operations. At the same time, the anthropomorphic arm hopes to move in a familiar way to help robots and humans coexist [8]. Traditional planning methods cannot be used for the above requirements. They can only plan a trajectory to achieve the simple task of mimicking a human arm functionally and improve performance during movement. However, they cannot Control the motion process in the attitude aspect. The problem of anthropomorphic arm movement planning has been studied by many scholars [9]-[11]. There are two schools of thought: the empirical statistical model school and the inverse kinematics solution school. The former first collects human measurement data from human experiments, and then USES regression model and other model tools to conduct statistical analysis on the data, forming attitude prediction model. The weakness of this school of thought is its lack of universality. In this way, it would be impractical to have to carry out a large number of experiments involving human bodies for each specific task.The latter usually USES biomechanics and kinematics as planning tools to solve joint trajectory by optimizing performance indexes and satisfying constraints. The deficiency of this method is the lack of experimental support, and many indicators are constructed by guessing. In recent years, more and more neurophysiological studies have shown that there are motion protectors in the movement of human arms [13]-[16].In the process of continuously completing a large number of tasks, human beings have formed a series of operational experience, which has evolved into a set of fixed motion mode, i.e movement primitives. In this way, one can easily assemble different motion primitives to perform difficult tasks such as building blocks, regardless of how each joint moves. Undoubtedly, this method provides a new way of thinking for motion planning, which is faced with a variety of business needs. By introducing the concept of anthropomorphic arm motion primitive, a new method of motion planning for complex tasks is proposed. The project The project is to design and model and do simulation of trajectory planning of anthropomorphic arm. Objective The manipulator can be regarded as a control system whose input and output are joint moment and endpoint position respectively. Starting from the theory of linear systems, the magnitude of the singular value of the output controllability matrix represents the influence of the input on the output [3] and [4]. Therefore, the singular vector corresponding to the maximum singular value represents the direction in which the joint torque (input) has the greatest influence on the end-effector position (output). This means that the singular vector also indicates the direction of maximum intensity of the perturbation effect with the endpoint position. Therefore, if we can plan the motion of the robot to properly control the direction of the singular vector, we can plan the direction of the endpoint position error caused by the disturbance. The direction of the singular vector is determined only by the dynamics of the robot and is therefore independent of the control input. Therefore, we can use a simple controller to control the direction of endpoint position error. Our goal is to model and design the trajectory planning of the anthropomorphic arm. In order to verify the effectiveness of our method, we will carry out simulation and modeling. Methodology The equation of motion of a n degree-of-freedom robotic arm can be described as a time-variant non-linear system, such as EMBED Equation.KSEE3 * MERGEFORMAT (1) where EMBED Equation.KSEE3 * MERGEFORMAT is the joint position, EMBED Equation.KSEE3 * MERGEFORMAT is the joint torque, EMBED Equation.KSEE3 * MERGEFORMAT is the inertia matrix, EMBED Equation.KSEE3 * MERGEFORMAT is the centrifugal force and the Coriolis force, and EMBED Equation.KSEE3 * MERGEFORMAT is the gravitational force acting on the robotic arm. Assuming that the Cartesian coordinate position of the end-effector is described as EMBED Equation.KSEE3 * MERGEFORMAT (m ≤ 3), the kinematic relation between θ and p can be described as a non-linear function such as EMBED Equation.KSEE3 * MERGEFORMAT (2) To apply the linear systems theory, we need to derive the linearized model of the robotic arm. Linearizing (1) and (2) with respect to the equilibrium points EMBED Equation.KSEE3 * MERGEFORMAT , EMBED Equation.KSEE3 * MERGEFORMAT , and EMBED Equation.KSEE3 * MERGEFORMAT , which satisfy EMBED Equation.KSEE3 * MERGEFORMAT , yields the linear time-invariant state equation and the output equation with n inputs, m outputs, and 2n state variables, as follows: EMBED Equation.KSEE3 * MERGEFORMAT (3) A system is said to be output controllable if it is possible to construct inputs that will transfer any given initial output to any final output until a finite time[6]. The output controllability matrix N of the robotic arm can be obtained by the matrices A, B, and C of (5). This yields EMBED Equation.KSEE3 * MERGEFORMAT (4) Than we reveal that the end-point position error given by the disturbance converges onto near the singular vector corresponding to the maximum singular value. We conduct example simulations using the two degree-of-freedom robotic arm in the horizontal plane as shown in Fig. 2. We define the joint torque and the end-point position (x, y) as the input 2008 IEEE/ASME International Conference on Advanced Intelligent Mechatronics, Xi'an, China, July, 2008. and output variables, respectively. Fig. 2. Two joint robotic arm which moves within a horizontal plane. If we can fit the direction of the singular vector to the tangential direction of the nominal trajectory, we will be able to obtain the robust trajectory from which the end- point suffered from the disturbance never deviate greatly. The following section describes the motion planning which enables the singular vector to point toward the tangential direction of the trajectory as much as possible. Development Environment This section describes the hardware and software environment we are going to use to develop the modelling and simulation of trajectory planning of anthropomorphic arm. Hardware Environment Computer 64GB, contains CPU and Memory. Software Environment We will be using Windows 10, Matlab, or other softwares we will do for us the job. CONCLUSION The present paper revealed that (1) the positional error of the end-point converged onto near the singular vector of the output controllability matrix, (2) the proposed trajectory generation algorithm enabled the singular vector to point toward the tangential direction of the trajectory, and (3) the end-point of the robotic arm could be controlled close to the nominal trajectory by using the proposed trajectory planning algorithm, even though the input torque includes the disturbance. Timeline NO Time periods Tasks 1 2019.01.01-2019.02.01 Research and study 2 2019.02.01-2019.02.15 Proposal report written 3 2019.02.15-2019.03.01 Literature review written 4 2019.03.01-2019.04.01 3D design and 2D design 5 2019.04.01-2019.04.15 Graduation thesis written References [1] J.E. Bobrow, S. Dubowsky, and J.S. Gibson,”Time-optimal Control of Robotic Manipulators Along Specified Paths,” Int. J. Robotics Research, vol.4, no. 3, pp. 3-17, 1985. [2] J.C. Doyle, K. Glover, P.P. Khargonekar, and B.A. Francis, ” State-Space Solutions to Standard H 2 and H ∞ control problem,” IEEE Trans. Automatic Control, vol. 34, no. 8, pp. 831-847, 1989. [3] B. Friedland, ”Controllability Index based on Conditioning Number,” Trans. of ASME, J. Dynamics Systems, Measurement and Control, pp. 444-445, December 1975. [4] C.D. Johnson, ”Optimization of a Certain Quality of Complete Controllability and Observability for Linear Dynamical Systems,” Trans. of ASME, J. Basic Engineering, pp.228-238, June 1969. [5] O. Khatib, ”Real-Time Obstacle Avoidance for Manipulators and Mobile Robots,” Int. J. Robotics Research, vol.5, no. 1, pp. 90-98, 1986. [6] E. Kreindler and P.E. Sarachik, ”On the Concepts of Controllability and Observability of Linear Systems,” IEEE Trans. on Automatic Control, vol. 9, no. 2, pp. 129-136, 1964. [7] J. Paattilammi and P.M. M#228;kil#228;, ”Fragility and Robustness: A case study on Paper Machine Headbox Control,” IEEE Control System Magazine, vol. 20, no. 1, pp.13-22, 2000. [8] E. Plaky and L.E. Kavraki, ”Distributed Sampling-based Roadmap of Trees for Large-scale Motion Planning,” Proc. of the 2005 IEEE International Conference on Robotics and Automation, pp. 3879-3884, 2005. [9] Z. Shiller and S. Dubowsky, ”On Computing the Global Time-optimal Motions of Robotic Manipulators in the Presence of Obstacle,” IEEE Trans. on Robotics and Automation, vol. 7, no. 6, pp. 785-797, 1991. [10] T. Yamawaki and M. Yashima, ”Effect of Gravity on Manipulation Performance of a Robotic Arm,” Proc. of the 2007 IEEE International Conference on Robotics and Automation, pp. 4407-4413, 2007. PAGE PAGE 2

2. 研究的基本内容、问题解决措施及方案

Trajectory planning of anthropomorphic arm Abstract In this paper, an algorithm for the direction control of endpoint position error caused by disturbance is proposed. With this in mind the effect of the disturbance on the position of the endpoint is maximized in the direction of the singular vector corresponding to the maximum singular value. Therefore, if we can control the direction of the singular vector, we can control the direction of the endpoint position error guided by the disturbance. The algorithm is applied to the hit motion of the manipulator, and the control algorithm and effect of the position error direction are presented. Trajectory planning of anthropomorphic arm Introduction In general, we cannot get the nominal model of real robot system, because real robot system contains various uncertainties. One uncertainty is noise and interference, which are excluded from the nominal model. For example, sensor noise and angular velocity quantization errors act as disturbances on the robot. In order to suppress disturbances, many scholars focus their research on the design of robust controllers [2] and [7]. The design of the controller is very complicated. At the same time, our concept is to effectively use the dynamics of the robot to complete the task of robustness, even if the controller is very simple. Therefore, we focus on motion planning rather than controller design. If the robust trajectory can be obtained independently of the control input, the controller can be simplified. In order to reduce the influence of disturbance on the task, we use the dynamic characteristics of the robot to plan the motion proactively. As far as we know, although the motion planning of [1], [9], [8] and other mechanical arms has been proposed, these schemes do not involve the problem of disturbance suppression. The results show that this method can effectively reduce the position error of the final point. On the other hand, this paper focuses on the direction of endpoint position error. In some tasks, the direction in which the position error is generated is more important than its size. One of these tasks is the strike motion of the robotic arm, as shown in figure 1. If the endpoint is disturbed and deviates vertically to the desired trajectory, the robot cannot hit the target, as shown in figure 1(a). If the direction of the position error can be controlled to be tangent to the desired trajectory, as shown in the figure. 1(b),Even with interference, the manipulator can hit the target. For hitting motion, there is no need to force the manipulator to reduce the size of position error. As long as the direction of position error can be controlled, the robustness of batting motion against disturbance can be improved. Therefore, this paper only deals with the generating direction of endpoint position error, and proposes a technology to control the generating direction to the tangential direction of the target path. Fig. 1. Application examples of the control of direction of the end-point position error caused by disturbance Related works In the traditional motion planning of industrial robots and redundant robots, the task is usually simple and fixed, and the planning is realized by optimizing predetermined performance indexes such as dexterity, joint limitation, torque limitation and obstacle avoidance With the gradual maturity of industrial manipulator technology, production efficiency is greatly improved, and the demand for robots in factories is also gradually reduced. Therefore, the research focus in the field of robotics has shifted from industrial robots [7] to service robots, and humanoid robots play a very important role in the field of service robots. Due to the various requirements faced by humanoid robots, the humanoid arm must be able to handle more complex task-level operations. At the same time, the anthropomorphic arm hopes to move in a familiar way to help robots and humans coexist [8]. Traditional planning methods cannot be used for the above requirements. They can only plan a trajectory to achieve the simple task of mimicking a human arm functionally and improve performance during movement. However, they cannot Control the motion process in the attitude aspect. The problem of anthropomorphic arm movement planning has been studied by many scholars [9]-[11]. There are two schools of thought: the empirical statistical model school and the inverse kinematics solution school. The former first collects human measurement data from human experiments, and then USES regression model and other model tools to conduct statistical analysis on the data, forming attitude prediction model. The weakness of this school of thought is its lack of universality. In this way, it would be impractical to have to carry out a large number of experiments involving human bodies for each specific task.The latter usually USES biomechanics and kinematics as planning tools to solve joint trajectory by optimizing performance indexes and satisfying constraints. The deficiency of this method is the lack of experimental support, and many indicators are constructed by guessing. In recent years, more and more neurophysiological studies have shown that there are motion protectors in the movement of human arms [13]-[16].In the process of continuously completing a large number of tasks, human beings have formed a series of operational experience, which has evolved into a set of fixed motion mode, i.e movement primitives. In this way, one can easily assemble different motion primitives to perform difficult tasks such as building blocks, regardless of how each joint moves. Undoubtedly, this method provides a new way of thinking for motion planning, which is faced with a variety of business needs. By introducing the concept of anthropomorphic arm motion primitive, a new method of motion planning for complex tasks is proposed. The project The project is to design and model and do simulation of trajectory planning of anthropomorphic arm. Objective The manipulator can be regarded as a control system whose input and output are joint moment and endpoint position respectively. Starting from the theory of linear systems, the magnitude of the singular value of the output controllability matrix represents the influence of the input on the output [3] and [4]. Therefore, the singular vector corresponding to the maximum singular value represents the direction in which the joint torque (input) has the greatest influence on the end-effector position (output). This means that the singular vector also indicates the direction of maximum intensity of the perturbation effect with the endpoint position. Therefore, if we can plan the motion of the robot to properly control the direction of the singular vector, we can plan the direction of the endpoint position error caused by the disturbance. The direction of the singular vector is determined only by the dynamics of the robot and is therefore independent of the control input. Therefore, we can use a simple controller to control the direction of endpoint position error. Our goal is to model and design the trajectory planning of the anthropomorphic arm. In order to verify the effectiveness of our method, we will carry out simulation and modeling. Methodology The equation of motion of a n degree-of-freedom robotic arm can be described as a time-variant non-linear system, such as EMBED Equation.KSEE3 * MERGEFORMAT (1) where EMBED Equation.KSEE3 * MERGEFORMAT is the joint position, EMBED Equation.KSEE3 * MERGEFORMAT is the joint torque, EMBED Equation.KSEE3 * MERGEFORMAT is the inertia matrix, EMBED Equation.KSEE3 * MERGEFORMAT is the centrifugal force and the Coriolis force, and EMBED Equation.KSEE3 * MERGEFORMAT is the gravitational force acting on the robotic arm. Assuming that the Cartesian coordinate position of the end-effector is described as EMBED Equation.KSEE3 * MERGEFORMAT (m ≤ 3), the kinematic relation between θ and p can be described as a non-linear function such as EMBED Equation.KSEE3 * MERGEFORMAT (2) To apply the linear systems theory, we need to derive the linearized model of the robotic arm. Linearizing (1) and (2) with respect to the equilibrium points EMBED Equation.KSEE3 * MERGEFORMAT , EMBED Equation.KSEE3 * MERGEFORMAT , and EMBED Equation.KSEE3 * MERGEFORMAT , which satisfy EMBED Equation.KSEE3 * MERGEFORMAT , yields the linear time-invariant state equation and the output equation with n inputs, m outputs, and 2n state variables, as follows: EMBED Equation.KSEE3 * MERGEFORMAT (3) A system is said to be output controllable if it is possible to construct inputs that will transfer any given initial output to any final output until a finite time[6]. The output controllability matrix N of the robotic arm can be obtained by the matrices A, B, and C of (5). This yields EMBED Equation.KSEE3 * MERGEFORMAT (4) Than we reveal that the end-point position error given by the disturbance converges onto near the singular vector corresponding to the maximum singular value. We conduct example simulations using the two degree-of-freedom robotic arm in the horizontal plane as shown in Fig. 2. We define the joint torque and the end-point position (x, y) as the input 2008 IEEE/ASME International Conference on Advanced Intelligent Mechatronics, Xi'an, China, July, 2008. and output variables, respectively. Fig. 2. Two joint robotic arm which moves within a horizontal plane. If we can fit the direction of the singular vector to the tangential direction of the nominal trajectory, we will be able to obtain the robust trajectory from which the end- point suffered from the disturbance never deviate greatly. The following section describes the motion planning which enables the singular vector to point toward the tangential direction of the trajectory as much as possible. Development Environment This section describes the hardware and software environment we are going to use to develop the modelling and simulation of trajectory planning of anthropomorphic arm. Hardware Environment Computer 64GB, contains CPU and Memory. Software Environment We will be using Windows 10, Matlab, or other softwares we will do for us the job. CONCLUSION The present paper revealed that (1) the positional error of the end-point converged onto near the singular vector of the output controllability matrix, (2) the proposed trajectory generation algorithm enabled the singular vector to point toward the tangential direction of the trajectory, and (3) the end-point of the robotic arm could be controlled close to the nominal trajectory by using the proposed trajectory planning algorithm, even though the input torque includes the disturbance. Timeline NO Time periods Tasks 1 2019.01.01-2019.02.01 Research and study 2 2019.02.01-2019.02.15 Proposal report written 3 2019.02.15-2019.03.01 Literature review written 4 2019.03.01-2019.04.01 3D design and 2D design 5 2019.04.01-2019.04.15 Graduation thesis written References [1] J.E. Bobrow, S. Dubowsky, and J.S. Gibson,”Time-optimal Control of Robotic Manipulators Along Specified Paths,” Int. J. Robotics Research, vol.4, no. 3, pp. 3-17, 1985. [2] J.C. Doyle, K. Glover, P.P. Khargonekar, and B.A. Francis, ” State-Space Solutions to Standard H 2 and H ∞ control problem,” IEEE Trans. Automatic Control, vol. 34, no. 8, pp. 831-847, 1989. [3] B. Friedland, ”Controllability Index based on Conditioning Number,” Trans. of ASME, J. Dynamics Systems, Measurement and Control, pp. 444-445, December 1975. [4] C.D. Johnson, ”Optimization of a Certain Quality of Complete Controllability and Observability for Linear Dynamical Systems,” Trans. of ASME, J. Basic Engineering, pp.228-238, June 1969. [5] O. Khatib, ”Real-Time Obstacle Avoidance for Manipulators and Mobile Robots,” Int. J. Robotics Research, vol.5, no. 1, pp. 90-98, 1986. [6] E. Kreindler and P.E. Sarachik, ”On the Concepts of Controllability and Observability of Linear Systems,” IEEE Trans. on Automatic Control, vol. 9, no. 2, pp. 129-136, 1964. [7] J. Paattilammi and P.M. M#228;kil#228;, ”Fragility and Robustness: A case study on Paper Machine Headbox Control,” IEEE Control System Magazine, vol. 20, no. 1, pp.13-22, 2000. [8] E. Plaky and L.E. Kavraki, ”Distributed Sampling-based Roadmap of Trees for Large-scale Motion Planning,” Proc. of the 2005 IEEE International Conference on Robotics and Automation, pp. 3879-3884, 2005. [9] Z. Shiller and S. Dubowsky, ”On Computing the Global Time-optimal Motions of Robotic Manipulators in the Presence of Obstacle,” IEEE Trans. on Robotics and Automation, vol. 7, no. 6, pp. 785-797, 1991. [10] T. Yamawaki and M. Yashima, ”Effect of Gravity on Manipulation Performance of a Robotic Arm,” Proc. of the 2007 IEEE International Conference on Robotics and Automation, pp. 4407-4413, 2007. PAGE PAGE 2