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We study doubly nonholonomic mobile manipulators composed of a nonholonomic mobile platform and a nonholonomic manipulator fixed to the platform. The kinematics of such mobile manipulators are represented by a pair of driftless control systems driven by platform and manipulator controls, and sharing an output function that describes positions and orientations of the end effector. The approach assumed in this paper is based on the concept of endogenous configuration defined as a pair of control functions of the platform, and of the aboard manipulator. For doubly nonholonomic mobile manipulators we define local performance measures and examine the inverse kinematic problem. Our results consist in a definition of the dexterity ellipsoid, a characterization of dextrous and isotropic endogenous configurations, and a derivation of a Jacobian pseudoinverse inverse kinematics algorithm. Computer simulations involving a 3 d.o.f. planar nonholonomic manipulator mounted on a kinematic car type platform illustrate our approach.