Modeling and Control in Task-Space of a Mobile Manipulator with Cancellation of Factory-Installed Proportional–Derivative Control
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Modeling and Control in Task-Space of a Mobile Manipulator with Cancellation of Factory-Installed Proportional–Derivative Control Gastón H. Salazar-Silva1, Marco A. Moreno-Armendáriz2, and Jaime Álvarez Gallegos3 1Unidad Profesional Interdisciplinaria en Ingeniería y Tecnologías Avanzadas, Instituto Politécnico Nacional, Mexico City, Mexico 2Centro de Investigación en Computación, Instituto Politécnico Nacional, Mexico City, Mexico 3Secretaría de Investigación y Posgrado, Instituto Politécnico Nacional, Mexico City, Mexico [email protected] Abstract. A mobile manipulator is a robotic arm modelado de manipuladores móvil que transforma el mounted on a mobile robot; a particular example is problema a el modelado de un manipulador estacionario a manipulator arm on a mobile robot with differential con restricciones cinemáticas no holónomas en traction. Mobile manipulators have many advantages las articulaciones. También se presenta un control over stationary manipulator, such as a larger work en el espacio de tarea que cancela un control space than a stationary manipulator could have in proporcional–diferencial proveniente de fabrica y usa practice. This paper shows a systematic approach un estimado de la derivada del modelo cinemático to modeling mobile manipulators that transforms the de postura. Finalmente, se presentan los resultados problem to the modeling of a stationary manipulator obtenidos en un experimento numérico. with non-holonomic kinematic constraints on the joints. Palabras clave. Control de robot, manipuladores It is also presented a task-space control that cancels móviles, cinemática de robots. a factory-installed proportional–derivative (PD) control and it uses an estimate of the derivative of the posture kinematic model. Finally, a numerical experiment is presented using this method. 1 Introduction Keywords. Robot control, mobile manipulators, robots The robots are coming out from the structured kinematics. environments in factories and they begin to appear in places such as houses, offices and hospitals, Modelado y control en espacio de where the surroundings have little structure or tarea de un manipulador móvil con not at all [8]; the mobile manipulators are a cancelación de control solution for these new work spaces. Basically, proporcional–derivativo instalado en a mobile manipulator is a stationary manipulator fábrica mounted on a mobile robot so the locomotion and manipulation tasks may be performed Resumen. Un manipulador móvil es un sistema simultaneously (Figure 1); these capabilities give to compuesto por un manipulador estacionario montado the mobile manipulator advantages over stationary sobre un robot móvil; un ejemplo particular es un brazo manipulador montado sobre un robot de tracción manipulators like a bigger task space and a diferencial. Los manipuladores móviles presentan varias greater autonomy; a mobile manipulator also has ventajas con respecto a manipuladores estacionarios, disadvantages, such as non-holonomic kinematic por ejemplo un mayor espacio de trabajo. En el constraints. Previously, the mobile-manipulator presente trabajo se muestra un método para el control was focused in handling separately the tasks of locomotion and manipulation, for example The present paper shows a new methodology in [17] the locomotion problem is the issue, or in for modeling and control a mobile manipulator [7, 19] where the manipulation was the control assuming that the robot behaves as an stationary problem; now these two tasks can be handled as manipulator with kinematic constraints on the one problem; for example, in [2, 17], a kinematic joints. The outline of this work is as follows: control is developed; there are literature that First a review on modeling techniques for presents results with the dynamic control of a stationary manipulators and mobile robots is mobile manipulator; for example in [1] a dynamic presented (Section 2). After that, an integrated control is used to reduce the balancing oscillation; modeling technique for kinematic models of also, a dynamic redundancy resolution control is mobile manipulators is presented (Section 3) and applied in [18] but the control is not robust; in [9] applied to obtain a dynamic model of the mobile a optimal dynamic control is developed to track a manipulator (Section 4). To obtain a kinematic trajectory considering the maximum load-carrying model the non-holonomic constraints are modeled capacity of the robot while it avoids obstacles. as a mapping between the actuation space and On the other hand, the kinematic modeling the joint space, using the so called configuration problem is still handled as two separate problems, kinematic model. Then, with the obtained and the resulting models are then combined to kinematic and dynamic models, a task-space obtain a model of the whole robot; these problem control is developed (Section 5). Finally, the results are solved using different techniques, for example of numerical simulations for the task-space control in [4, 10, 15, 14, 13, 3]; in [12] a method is are showed assuming a 5-DOF differential-traction presented that combines the kinematic models mobile manipulator (Section 6). of the mobile base and the manipulator, but the mobile base and the manipulator are still modeled 2 Kinematic Modeling Techniques for with different methods; very interesting examples are [14] and [13], where both the mobile base and Stationary and Mobile Robots the manipulator have non-holonomic constraints and yet are modeled by different methods. A In a stationary manipulator, a kinematic model fundamental work for mobile-robots modeling is describes the relation between the motion of [5]; in it a classification for wheeled mobile robots a mechanical system and the motion of the is presented, based on their degree of mobility actuators. This model is conceptually obtained and degree of maneuverability, and four types of through the so called forward kinematics, which is models are proposed. the function that describes the relation between the posture of the final effector of the robot in task space and the value of the joint variables of the robot. The forward kinematics is usually expressed as rm = fm(q) (1) p where rm 2 R is the posture variables vector, n q 2 R is the joints displacement vector, and n is the dimension of q, usually called degree of freedom (DOF). A widely used tool to obtain the forward kinematics of a stationary manipulator is the homogeneous transformation matrix [16]; this matrix can be constructed with the help of the Denavit–Hartenberg parameters; these parameters describes the geometric characteristics of the links and joints of a robot. The kinematic model of a stationary manipulator with full-actuated independent joints is then Figure 1. The experimental system is composed of a obtained through the time derivative of (1) by a Pioneer 3DX mobile robot and a Cyton manipulator arm with 7 DOF r˙m(t) = Jm(q)q˙(t) (2) p n where r˙m 2 R are the posture velocities, q˙ 2 R where x and y are the coordinates on the plane on are the joint velocities of the manipulator, and the which the mobile robot moves and f is the robot p×n matrix Jm(q) 2 R is the so called Jacobian and it orientation on such plane. If the mobile robot has is defined as centered orientable wheels, the posture must be ¶ fm extended to include the angles of the wheels. The Jm(q) = (q). ¶q posture kinematic model of a wheeled mobile robot Another way to compute the Jacobian is the with differential traction can be expressed as [5] geometric method [16], that has the advantage of not requiring an explicit computation of the r˙b(t) = B(q)hb (4) derivatives of the forward kinematic, as uses n−k where hb(t) 2 R is the vector which contains numeric information from the homogeneous n×(n−k) transformations. the velocities of the actuators, and B(qb) 2 R is a matrix with its columns being a base of On the other hand, the motion of a wheeled the null space of the non-holonomic constraints; mobile robot is characterized by the kinematic The posture kinematic model is useful in the constraints imposed by the wheels [11, 5]. computation of control laws in the task space; on A kinematic constraint may be holonomic or the other hand, the configuration kinematic model non-holonomic. A mechanical system is said to be is used to simplify the dynamic model of mobile holonomic if there is a set of k constraints to the robot. motion; these constraints may be expressed as The second model describes a relation between hi(q) = 0, i = 1, :::,k the joints motion and the actuators motion, called configuration kinematic model, and it is defined as where hi are scalar functions; such constraints are geometric and they limit where may be the system q˙b = Sb(q)hb (5) configuration. If the mechanical system is limited n×(n−k) by constraints expressed as where Sb(q) 2 R is a matrix with its columns belongs to the null space of the constraint matrix ai(q,q ˙) = 0 Ab. For example, in a differential-traction mobile then the system is called non-holonomic and these robot, the configuration kinematic model can be constraints limit how the mechanical system can defined as move, but not restrict where the system can be. 0 cosf 0 1 A special kind of non-holonomic constraint is the S (q) = sinf 0 . so the called Pfaffian form, in where the constraint b @ A 0 1 equations are linear with respect the joint velocities It is important to remark that S (q) is an annihilator ai(q)q˙ = 0. b of the kinematic constraints, such that The kinematic model of a mobile robot can be T obtained from the null space of the non-holonomic Ab(q) Sb(q) = 0 (6) kinematic constraints of the robot. n×k In mobile robots, there are two kinds of kinematic where the matrix Ab(q) 2 R defines the models [11, 5].