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We present a motion synthesis framework capable of producing a wide variety of important human behaviors that have rarely been studied, including getting up from the ground, crawling, climbing, moving heavy objects, acrobatics (hand-stands in particular), and various cooperative actions involving two characters and their manipulation of the environment. Our framework is not specific to humans, but applies to characters of arbitrary morphology and limb configuration. The approach is fully automatic and does not require domain knowledge specific to each behavior. It also does not require pre-existing examples or motion capture data. At the core of our framework is the contact-invariant optimization (CIO) method we introduce here. It enables simultaneous optimization of contact and behavior. This is done by augmenting the search space with scalar variables that indicate whether a potential contact should be active in a given phase of the movement. These auxiliary variables affect not only the cost function but also the dynamics (by enabling and disabling contact forces), and are optimized together with the movement trajectory. Additional innovations include a continuation scheme allowing helper forces at the potential contacts rather than the torso, as well as a feature-based model of physics which is particularly well-suited to the CIO framework. We expect that CIO can also be used with a full physics model, but leave that extension for future work.