The main goal of this thesis is to simulate the response of a humanoid robot using a specified control algorithm which can achieve a sustainable biped locomotion with 4 basic locomotion phases. Basic parts for the body of the humanoid robot model are shaped according to the specified basic physical parameters and assumed kinematic model. The kinematic model, which does not change according to locomotion phases and consists of 27 segments including 14 virtual segments, provides a humanoid robot model with 26 degrees of freedom (DOF). Corresponding kinematic relations for the robot model are obtained by recursive formulations. Derivation of dynamic equations is carried out by the Newton-Euler formulation. A trajectory definition algorithm which defines positions, orientations, translational and angular velocities for the hip and its mass center, toe part of the foot and its toe point is created. A control strategy based on predictive optimum command acceleration calculations and computed torque control method is implemented. The simulation is executed in Simulink and the visualization of the simulation is established in a virtual environment by Virtual Reality Toolbox of MATLAB. The simulation results and the user defined reference input are displayed simultaneously in the virtual environment. In this study, a simulation environment for the biped locomotion of humanoid robots is created. By the help of this thesis, the user can test various control strategies by modifying the modular structure of the simulation and acquire necessary information for the preliminary design study of a humanoid robot construction.

During fast point-to-point motion, the inherent joint flexibility could be detrimental in terms of residual vibration. Aiming to minimize the vibration, the command shaping method has been developed so as to remove critical energy from the input profile at resonant frequencies. Since this method requires information of a physical model in order to find the target frequencies, the quality of the shaped command profile relies on the accuracy of the model parameter estimation. Therefore, in this work, a system identification method using Instrumental Variables is applied from the literature. Compared with the classic Ordinary Least Square method, the IV approach has successfully improved the estimation of parameters, based on simulation results. The accuracy of parameter estimation influences the command profile, as does the feedback controller. In this work, starting from a mathematical derivation with a mismatch model due to a feedback controller called Computed Torque Control, insight for the closed-loop system is given with regard to the interaction between control gains and the actual resonant frequencies. It is found that the control gain is able to modify the actual resonant frequency curve, and push it into or out of the shaping bounds which are generated from the command shaping method. Further analysis based on the simulation results shows that the overlap area between the shaping bounds and the actual frequencies affects the level of residual vibration. In light of this fact, an optimal control gain exists and is found when the estimation error is in a certain range. At the end, recommendations for choosing the control gains are provided.

This thesis presents scaling methods for sizing a prototype micro prismatic revolute (PR) manipulator actuated by permanent magnet (PM) direct current (d.c.) gearmotors. Dimensional analysis was the principle tool used in this investigation, and addressed the problems of scaling a trajectory planner, control law, and gearmotors that exhibit internal nonlinear friction. Similitude methods were used to develop a scaleable two degree-of-freedom trajectory planner from a third order polynomial. Scaling laws were developed from Buckingham's Pi theorem to facilitate the selection process of gearmotors. Nondimensional, nonlinear, differential equations were developed to describe viscous, Coulomb and static friction in comparative PM d.c. motors. From the insights gained through dimensional analysis, a scaleable controller based on the computed torque method was developed and implemented with a cubic trajectory planner. Model and prototype PR manipulator systems were simulated using a hybrid Matlab/Simulink simulation scheme. Experimental systems were constructed with dissimilar model and prototype motors. Control was provided by an AT class PC equipped with 12-bit A/D, D/A cards operating at a sample rate of 100 Hz. The control algorithm was written in Borland 3.1 C for DOS. Results from the experimental testing showed excellent agreement between the test and simulated data and verified the viability of the scaling laws. The techniques presented in this thesis are expected to be applicable to any application that involves scaling PM d.c. micro gearmotors that have significant internal friction terms. These simple, practical tools should be especially beneficial to designers of micro robotic systems. / Master of Science

Este trabalho apresenta uma comparação entre as estratégias de controle por torque calculado e controle repetitivo aplicadas a manipuladores robóticos. O objetivo no uso desses controladores é para que o manipulador siga referência de trajetória periódica no espaço das juntas. O desenvolvimento e implementação dos controladores são focados no manipulador WAM (Whole Arm Manipulator) da Barrett Technology®Inc. Neste trabalho, também são apresentadas uma formulação do modelo não linear do manipulador e as sínteses dos controladores por torque calculado e repetitivo aplicados ao modelo do manipulador linearizado por realimentação. O controlador por torque calculado é apresentado e sintetizado na sua forma clássica. Para o controlador repetitivo, a síntese parte do princípio do modelo interno com a adição de uma estrutura repetitiva e uma realimentação proporcional e derivativa do erro de seguimento de referência O projeto dos ganhos do controlador repetitivo é feito através de um problema de otimização convexa com restrições na forma de inequações matriciais lineares (ou no inglês: Linear Matrix Inequalities - LMI). A formulação do problema de otimização parte da teoria de estabilidade segundo Lyapunov com um funcional Lyapunov-Krasoviskii, adição de um custo quadrático, para ajuste de desempenho, e de um critério de desempenho transitório dado pela taxa de decaimento exponencial da norma dos estados. É apresentada a comparação entre as estratégias de controle e a validação do controlador repetitivo proposto aplicado ao caso com linearização perfeita e ao caso com o modelo não linear do manipulador. No primeiro caso, é feita a simulação do modelo linear do manipulador com adição de um torque de atrito na junta. No segundo caso, é utilizado o sistema ROS (Robot Operating System) com o programa Gazebo simulando o manipulador WAM considerando erros de linearização, isto é, incertezas paramétricas. / This work presents a comparison between the strategies of computed-torque control and repetitive control applied to robotic manipulators. The main objective in use these controllers with the manipulator is to tracking periodic trajectory in joint space. The development and implementation of controllers are focused on the Whole Arm Manipulator (WAM) of the Barrett Technology®Inc. Also featured are a non-linear model formulation of the manipulator and the synthesis of controllers for computed-torque control and repetitive control applied to the manipulator model linearized by state feedback. The computed-torque controller is presented in its classic form. For the repetitive controller, the synthesis is based on the internal model principle with the addition of a repetitive structure and a proportional-derivative reference tracking error feedback. The design of the repetitive controller gains is done through a convex optimization problem with linear matrix inequalities (LMI) constraints. The formulation of the optimization problem is based on the Lyapunov stability theory using a Lyapunov-Krasoviskii functional, addition of a quadratic cost for performance adjustment and a transient performance criteria given by the exponential decay rate of the states norm. A comparison between the control strategies and the validation of the repetitive controller applied to the case with perfect linearization and the case with the non-linear model of the manipulator are presented. In the first case, is made simulations of the linear model of the manipulator in MATLAB program, with the addition of a disturbance modeling the friction torque at the joint. In the second case, is used the Robot Operating System (ROS) with Gazebo program simulating the WAM nonlinear model. In this case, a possible mismatch between the model used for the feedback linearization and the real system is taken into account.

Este trabalho apresenta uma comparação entre as estratégias de controle por torque calculado e controle repetitivo aplicadas a manipuladores robóticos. O objetivo no uso desses controladores é para que o manipulador siga referência de trajetória periódica no espaço das juntas. O desenvolvimento e implementação dos controladores são focados no manipulador WAM (Whole Arm Manipulator) da Barrett Technology®Inc. Neste trabalho, também são apresentadas uma formulação do modelo não linear do manipulador e as sínteses dos controladores por torque calculado e repetitivo aplicados ao modelo do manipulador linearizado por realimentação. O controlador por torque calculado é apresentado e sintetizado na sua forma clássica. Para o controlador repetitivo, a síntese parte do princípio do modelo interno com a adição de uma estrutura repetitiva e uma realimentação proporcional e derivativa do erro de seguimento de referência O projeto dos ganhos do controlador repetitivo é feito através de um problema de otimização convexa com restrições na forma de inequações matriciais lineares (ou no inglês: Linear Matrix Inequalities - LMI). A formulação do problema de otimização parte da teoria de estabilidade segundo Lyapunov com um funcional Lyapunov-Krasoviskii, adição de um custo quadrático, para ajuste de desempenho, e de um critério de desempenho transitório dado pela taxa de decaimento exponencial da norma dos estados. É apresentada a comparação entre as estratégias de controle e a validação do controlador repetitivo proposto aplicado ao caso com linearização perfeita e ao caso com o modelo não linear do manipulador. No primeiro caso, é feita a simulação do modelo linear do manipulador com adição de um torque de atrito na junta. No segundo caso, é utilizado o sistema ROS (Robot Operating System) com o programa Gazebo simulando o manipulador WAM considerando erros de linearização, isto é, incertezas paramétricas. / This work presents a comparison between the strategies of computed-torque control and repetitive control applied to robotic manipulators. The main objective in use these controllers with the manipulator is to tracking periodic trajectory in joint space. The development and implementation of controllers are focused on the Whole Arm Manipulator (WAM) of the Barrett Technology®Inc. Also featured are a non-linear model formulation of the manipulator and the synthesis of controllers for computed-torque control and repetitive control applied to the manipulator model linearized by state feedback. The computed-torque controller is presented in its classic form. For the repetitive controller, the synthesis is based on the internal model principle with the addition of a repetitive structure and a proportional-derivative reference tracking error feedback. The design of the repetitive controller gains is done through a convex optimization problem with linear matrix inequalities (LMI) constraints. The formulation of the optimization problem is based on the Lyapunov stability theory using a Lyapunov-Krasoviskii functional, addition of a quadratic cost for performance adjustment and a transient performance criteria given by the exponential decay rate of the states norm. A comparison between the control strategies and the validation of the repetitive controller applied to the case with perfect linearization and the case with the non-linear model of the manipulator are presented. In the first case, is made simulations of the linear model of the manipulator in MATLAB program, with the addition of a disturbance modeling the friction torque at the joint. In the second case, is used the Robot Operating System (ROS) with Gazebo program simulating the WAM nonlinear model. In this case, a possible mismatch between the model used for the feedback linearization and the real system is taken into account.

Este trabalho apresenta uma comparação entre as estratégias de controle por torque calculado e controle repetitivo aplicadas a manipuladores robóticos. O objetivo no uso desses controladores é para que o manipulador siga referência de trajetória periódica no espaço das juntas. O desenvolvimento e implementação dos controladores são focados no manipulador WAM (Whole Arm Manipulator) da Barrett Technology®Inc. Neste trabalho, também são apresentadas uma formulação do modelo não linear do manipulador e as sínteses dos controladores por torque calculado e repetitivo aplicados ao modelo do manipulador linearizado por realimentação. O controlador por torque calculado é apresentado e sintetizado na sua forma clássica. Para o controlador repetitivo, a síntese parte do princípio do modelo interno com a adição de uma estrutura repetitiva e uma realimentação proporcional e derivativa do erro de seguimento de referência O projeto dos ganhos do controlador repetitivo é feito através de um problema de otimização convexa com restrições na forma de inequações matriciais lineares (ou no inglês: Linear Matrix Inequalities - LMI). A formulação do problema de otimização parte da teoria de estabilidade segundo Lyapunov com um funcional Lyapunov-Krasoviskii, adição de um custo quadrático, para ajuste de desempenho, e de um critério de desempenho transitório dado pela taxa de decaimento exponencial da norma dos estados. É apresentada a comparação entre as estratégias de controle e a validação do controlador repetitivo proposto aplicado ao caso com linearização perfeita e ao caso com o modelo não linear do manipulador. No primeiro caso, é feita a simulação do modelo linear do manipulador com adição de um torque de atrito na junta. No segundo caso, é utilizado o sistema ROS (Robot Operating System) com o programa Gazebo simulando o manipulador WAM considerando erros de linearização, isto é, incertezas paramétricas. / This work presents a comparison between the strategies of computed-torque control and repetitive control applied to robotic manipulators. The main objective in use these controllers with the manipulator is to tracking periodic trajectory in joint space. The development and implementation of controllers are focused on the Whole Arm Manipulator (WAM) of the Barrett Technology®Inc. Also featured are a non-linear model formulation of the manipulator and the synthesis of controllers for computed-torque control and repetitive control applied to the manipulator model linearized by state feedback. The computed-torque controller is presented in its classic form. For the repetitive controller, the synthesis is based on the internal model principle with the addition of a repetitive structure and a proportional-derivative reference tracking error feedback. The design of the repetitive controller gains is done through a convex optimization problem with linear matrix inequalities (LMI) constraints. The formulation of the optimization problem is based on the Lyapunov stability theory using a Lyapunov-Krasoviskii functional, addition of a quadratic cost for performance adjustment and a transient performance criteria given by the exponential decay rate of the states norm. A comparison between the control strategies and the validation of the repetitive controller applied to the case with perfect linearization and the case with the non-linear model of the manipulator are presented. In the first case, is made simulations of the linear model of the manipulator in MATLAB program, with the addition of a disturbance modeling the friction torque at the joint. In the second case, is used the Robot Operating System (ROS) with Gazebo program simulating the WAM nonlinear model. In this case, a possible mismatch between the model used for the feedback linearization and the real system is taken into account.

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