1 Modelling of tree structures 変更行列モデル・models 運動学 微分運動学 力学連続チェインと同じ Modelling of closed chain  特性：  Enhanced accuracy and load-carrying capacity of the robot.  L 関節 n + 1 リンク L>n  Independent closed loops number is: B=L–n  Joints are either active or passive N active joints  mj=1 if joint actuated  mj=0 if joint non-actuated Modelling of closed chain ① Equivalent tree structure with n joints by virtually cutting each closed chain at one of its passive joints. Geometric parameters of the equivalent tree defined as before ② Cut joints numbered n + 1 to L. Each cut joint k, attach Rk fixed on one of the links connected to this joint, link j. The zk axis is taken along the axis of joint k, and the xk axis is aligned with the common normal between zk and zj ③ Rk fixed on link j, transformation matrix between Rj and Rk constant. For clearness, this transformation will be denoted jTk+B, with j = a(k+B). The geometric parameters defining this transformation have a subscript k + B. Note that frame Rk+B is aligned with frame Rk, and that both rk+B and θk+B are zero. Geometric description of a structure with closed loops is defined by an equivalent tree structure that is obtained by cutting each closed loop at one of its joints and by adding two frames at each cut joint. The total number of frames is equal to n + 2B and the geometric parameters of the last B frames are constants.