The Cauchy stress equations (1823), the Cosserat couple-stress equations (1909), the Clausius virial equation (1870) and the Maxwell/Weyl equations (1873, 1918) are among the most famous partial differential equations...The Cauchy stress equations (1823), the Cosserat couple-stress equations (1909), the Clausius virial equation (1870) and the Maxwell/Weyl equations (1873, 1918) are among the most famous partial differential equations that can be found today in any textbook dealing with elasticity theory, continuum mechanics, thermodynamics or electromagnetism. Over a manifold of dimension n, their respective numbers are n,n(n−1)/2,1,nwith a total of N=(n+1)(n+2)/2, that is 15 when n=4for space-time. This is also just the number of parameters of the Lie group of conformal transformations with n translations, n(n−1)/2rotations, 1 dilatation and n highly non-linear elations introduced by E. Cartan in 1922. The purpose of this paper is to prove that the form of these equations only depends on the structure of the conformal group for an arbitrary n≥1because they are described as a whole by the (formal) adjoint of the first Spencer operator existing in the Spencer differential sequence. Such a group theoretical implication is obtained by applying totally new differential geometric methods in field theory. In particular, when n=4, the main idea is not to shrink the group from 10 down to 4 or 2 parameters by using the Schwarzschild or Kerr metrics instead of the Minkowski metric, but to enlarge the group from 10 up to 11 or 15 parameters by using the Weyl or conformal group instead of the Poincaré group of space-time. Contrary to the Einstein equations, these equations can be all parametrized by the adjoint of the second Spencer operator through Nn(n−1)/2potentials. These results bring the need to revisit the mathematical foundations of both General Relativity and Gauge Theory according to a clever but rarely quoted paper of H. Poincaré (1901). They strengthen the recent comments we already made about the dual confusions made by Einstein (1915) while following Beltrami (1892), both using the same Einstein operator but ignoring it is self-adjoint in the framework of differential double duality.展开更多
为避免机器人运动学参数辨识过程中,测量坐标系与机器人基坐标系之间繁琐的坐标变换,首先利用关节旋量的空间几何特性,提出了基于伴随变换的距离误差模型。其次,针对距离误差模型中可辨识参数的冗余性,通过辨识雅可比矩阵的零空间分析,...为避免机器人运动学参数辨识过程中,测量坐标系与机器人基坐标系之间繁琐的坐标变换,首先利用关节旋量的空间几何特性,提出了基于伴随变换的距离误差模型。其次,针对距离误差模型中可辨识参数的冗余性,通过辨识雅可比矩阵的零空间分析,确定了可辨识参数的数目与误差测量方式之间的关系。确定了绕对应关节旋转的测量方式和相对初始位形的测量方式下可辨识参数的数目。最后,对KUKA you Bot机器人的运动学参数辨识进行了实验研究,实验结果验证了距离误差模型的有效性和参数冗余性分析的正确性。展开更多
文摘The Cauchy stress equations (1823), the Cosserat couple-stress equations (1909), the Clausius virial equation (1870) and the Maxwell/Weyl equations (1873, 1918) are among the most famous partial differential equations that can be found today in any textbook dealing with elasticity theory, continuum mechanics, thermodynamics or electromagnetism. Over a manifold of dimension n, their respective numbers are n,n(n−1)/2,1,nwith a total of N=(n+1)(n+2)/2, that is 15 when n=4for space-time. This is also just the number of parameters of the Lie group of conformal transformations with n translations, n(n−1)/2rotations, 1 dilatation and n highly non-linear elations introduced by E. Cartan in 1922. The purpose of this paper is to prove that the form of these equations only depends on the structure of the conformal group for an arbitrary n≥1because they are described as a whole by the (formal) adjoint of the first Spencer operator existing in the Spencer differential sequence. Such a group theoretical implication is obtained by applying totally new differential geometric methods in field theory. In particular, when n=4, the main idea is not to shrink the group from 10 down to 4 or 2 parameters by using the Schwarzschild or Kerr metrics instead of the Minkowski metric, but to enlarge the group from 10 up to 11 or 15 parameters by using the Weyl or conformal group instead of the Poincaré group of space-time. Contrary to the Einstein equations, these equations can be all parametrized by the adjoint of the second Spencer operator through Nn(n−1)/2potentials. These results bring the need to revisit the mathematical foundations of both General Relativity and Gauge Theory according to a clever but rarely quoted paper of H. Poincaré (1901). They strengthen the recent comments we already made about the dual confusions made by Einstein (1915) while following Beltrami (1892), both using the same Einstein operator but ignoring it is self-adjoint in the framework of differential double duality.
文摘为避免机器人运动学参数辨识过程中,测量坐标系与机器人基坐标系之间繁琐的坐标变换,首先利用关节旋量的空间几何特性,提出了基于伴随变换的距离误差模型。其次,针对距离误差模型中可辨识参数的冗余性,通过辨识雅可比矩阵的零空间分析,确定了可辨识参数的数目与误差测量方式之间的关系。确定了绕对应关节旋转的测量方式和相对初始位形的测量方式下可辨识参数的数目。最后,对KUKA you Bot机器人的运动学参数辨识进行了实验研究,实验结果验证了距离误差模型的有效性和参数冗余性分析的正确性。
