Matsuno and Saito (2001) have proposed an attitude control law, considering a planar two-link space robot with initial angular momentum, i.e., a typical example of a 3-state and 2-input affine system with a drift term. All the previous research was based on the assumption that the system is at rest initially, i.e., the initial momentum of FFSMS is 0. Mathematically speaking, an FFSMS with initial angular momentum is an affine system with a drift term. Umetani and Yoshida (1989) introduced the free-floating system generalized Jacobian matrix (GJM) and developed a resolved rate and acceleration control method based on it. The matrix R0(ε,n) is the rotation matrix between the spacecraft 0th frame and the inertial frame, expressed as a function of the spacecraft Euler parameters ε, n. The 3 × 3 matrix 0D is the inertia matrix of the entire system as seen from the system CM, expressed in the spacecraft 0th frame, and as such it is a positive definite symmetric matrix and thus always invertible.
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0ω0 is the spacecraft angular velocity, expressed in the spacecraft 0th frame and the N × 1 column-vectors q,q˙ represent manipulator joint angles and rates, respectively. 6 DOF in total including the DOF of the spacecraft. 35 is 0 for any possible choice of the matrix C(q,q˙), since it results from the principle of the energy conservation (Xu and Shum, 1991). Examining the second term of the RHS of Eq. The matrix H is an N × N symmetric and positive definite matrix called the reduced system inertial matrix and defined in Appendix A, and the N × N matrix C(q,q˙) contains the non-linear Coriolis and centrifugal terms for a spatial FFSMS with zero angular momentum and can be written in several forms. Yamada et al. (1995) have presented a path planning scheme for a single arm of an FFSMS, which is equipped with momentum wheels.
This term is caused by the angular momentum and complicates the path planning and control of such systems. In this section, the differential kinematics and the equations of motion of a rigid FFSMS with NZAM are briefly developed. It is shown that despite the presence of NZAM, these structural properties are still valid. Xu and Shum (1991) derived the equations of motion of FFSMS with zero angular momentum in joint and inertial space, applying the Langragian methodology. However, in Nanos and Papadopoulos (2015), a methodology to avoid such singularities, by carefully choosing the FFSMS initial configuration, has been developed. болка в горната част на китката
. Thus, Eq. 9 can be used to derive the joint angles required so that the end-effector follows a desired path, described by the end-effector velocity vE. Whether this will happen, depends on the path taken by the end-effector. Small spacecraft equipped with robotic manipulators will be the backbone of any robotic on-orbit servicing, asteroid mining, or active space debris removal missions.
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In this case, the end-effector velocity r˙E, as the vector sum of these velocities, will have the same direction regardless the joint rates and the end-effector can move only along this direction. T is the manipulator torque vector where τi is the torque applied on the ith joint. 15, are transformed here to the Cartesian space. 35 are 0, and the property described by Eq. 14 on the spacecraft’s attitude, described by the Euler parameters ε, n. Note that the additional terms Ch and gh, caused by the presence of initial angular momentum, are functions of the spacecraft attitude described by the Euler parameters ε, n. Thus, in the spatial case, the system’s reduced equations of motion depend on the spacecraft’s attitude.
This method utilizes the angular momentum of the base, without causing its nutation, which occurs unless the final attitude of the base is the same as the initial one. More recently, a trajectory optimization method for systems with non-conserved linear and angular momentum has been proposed (Rybus et al., 2016). Nanos and Papadopoulos (2011) have proposed an approach that identifies workspace areas and required joint motions so that the end-effector can remain fixed despite the NZAM. The arm kinematics were validated using the method of the capability map. The velocities, u¯, and manipulator joint displacements, θm, can be propagated using state transition matrices. Papadopoulos and Dubowsky (1991a), based on the similarities of the structure of the kinematics and dynamics between fixed base and FFSMS with zero angular momentum, showed that almost any terrestrial fixed base control algorithm can be applied to FFSMS with zero angular momentum, considering some additional conditions.