动基座刚性机器人动力学的有效几何线性化
Efficient Geometric Linearization of Moving-Base Rigid Robot Dynamics
摘要：
机器人系统关于给定状态输入轨迹（包括受控平衡状态）的运动方程的线性化对于基于模型的规划、闭环控制、增益调整和状态估计是有价值的工具。与具有棱柱或旋转关节的固定基座机械手的情况相反，移动基座机器人系统（诸如类人机器人、四足机器人或空中机械手）的状态空间不能由有限数量的独立坐标全局参数化。这种不可能性是这样一个事实的直接结果，即这些系统的状态包括系统的全局定向，形式上描述为特殊正交群SO（3）的一个元素。因此，从实际的角度来看，获得这些系统的运动方程的线性化通常是通过借助于例如，欧拉角或卡丹角。然而，这有引入人工参数化奇点和额外的导数计算的缺点。在这方面的贡献，我们表明，它实际上是可以定义一个概念的线性化，不需要使用的局部参数化系统的方向，获得一个数学上优雅的，递归的，和无奇异线性化移动为基础的机器人系统。递归，特别是，通过提出一个非平凡的修改现有的递归算法，以允许计算的几何导数的逆动力学和逆的机器人系统的质量矩阵。通过与几何有限差分结果的数值比较，验证了算法的正确性。
英文摘要：
The linearization of the equations of motion of a robotics system about a given state-input trajectory, including a controlled equilibrium state, is a valuable tool for model-based planning, closed-loop control, gain tuning, and state estimation. Contrary to the case of fixed based manipulators with prismatic or rotary joints, the state space of moving-base robotic systems such as humanoids, quadruped robots, or aerial manipulators cannot be globally parametrized by a finite number of independent coordinates. This impossibility is a direct consequence of the fact that the state of these systems includes the system's global orientation, formally described as an element of the special orthogonal group SO(3). As a consequence, obtaining the linearization of the equations of motion for these systems is typically resolved, from a practical perspective, by locally parameterizing the system's attitude by means of, e.g., Euler or Cardan angles. This has the drawback, however, of introducing artificial parameterization singularities and extra derivative computations. In this contribution, we show that it is actually possible to define a notion of linearization that does not require the use of a local parameterization for the system's orientation, obtaining a mathematically elegant, recursive, and singularity-free linearization for moving-based robot systems. Recursiveness, in particular, is obtained by proposing a nontrivial modification of existing recursive algorithms to allow for computations of the geometric derivatives of the inverse dynamics and the inverse of the mass matrix of the robotic system. The correctness of the proposed algorithm is validated by means of a numerical comparison with the result obtained via geometric finite difference.
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