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This work deals with force-controlled robot tasks under uncertainty. The aim is to increase the robot’s autonomy in order to perform compliant motion tasks in unstructured environments. A general autonomous compliant motion system consists of three components, i.e., the task planner, the force controller and the estimator. This book focuses on solutions where these components are based on the same geometrical contact models. The models specify the constraints, the motion degree of freedom directions and the ideal contact force directions for every contact formation (CF). The models are a function of the inaccurately known geometrical parameters, i.e., the positions, orientations and dimensions of the contacting objects. During the compliant motion, the parameters are estimated more accurately based on pose, twist and wrench measurements.

This work describes the latest contributions in the areas of fast recursive estimation, contact modelling and task planning with active sensing. The next section presents a detailed overview of those contributions.

The Linear Regression Kalman Filter (LRKF, Sect. 4.2) has the following properties:

This appendix contains the derivation of the process and measurement update of this LRKF (Sects. A.2 and A.3). First, Sect. A.1 describes the linear regression formulas.

This appendix contains the proofs of Theorems 5.1 and 5.2 describing the Non-minimal State Kalman Filter (NMSKF). The interpretation of the filter and examples are presented in Chap. 5.

Chapter 6 describes the wrench spanning sets of the elementary contact library with respect to a frame {c} on the contact. In order to apply the measurement equations, the twist and wrench measurements and the wrench spanning set need to be expressed with respect to the same frame.

This appendix presents the analytical expressions for the wrench screw transformation matrices , the rotation matrices , the skew-symmetric matrices and the homogeneous transformation matrices .

This appendix contains the proof that the Kalman Filter (KF) is robust against non-minimal measurement equations. The KF algorithm uses the inverse of the innovation covariance matrix . For non-minimal measurement equations, this matrix is singular. This appendix contains the proof that the results of the Kalman Filter (KF) using non-minimal measurement equations are the same as the results of the KF using a minimal set of measurement equations, .

This appendix contains the proofs of Lemmas 7.1 and 7.2. These lemmas state that, for systems, (i) a Kalman Filter can be run on only the state variables; and (ii) the full state estimate and covariance matrix (i.e., including the state variables) can be calculated at any time based on the full initial state estimate and covariance matrix and the new state estimate and covariance matrix of the observed part of the state.

The measurement equations of a multiple-contact CF can be generated from the wrench spanning sets and closure equations of a number () of elementary CFs. This appendix describes how for each of these elementary contacts (1≤ ≤ ) a state vector ′ can be found which linearizes both the wrench spanning set (′,) and the closure equation (′,) = 0. Section 7.3.2 uses these ′ to write quasi-linear contact models for the Nonminimal State Kalman Filter (NMSKF).

This appendix describes how to calculate a spanning set for the CF-observable parameter space for twist and pose measurements.