# Robot learning by demonstration/Correspodence Figure 1: Robots and Humans, while inhabiting the same space and interacting with the same objects, and perhaps even superficially similar, still perceive and interact with the world in fundamentally different ways. Credit: Basilio Noris, LASA-EPFL

To evaluate the similarity between the human and robot behaviors, we must first deal with the fact that the human and the robot may occupy different state spaces, of perhaps different dimensionalities. Generally, this is always the case unless immersive teleoperation is used to generate demonstrations. Thus, as a first step towards computing $$M\ ,$$ we must identify correspondences between the state spaces as illustrated in #correspond. We identify three different ways in which states $$\mathbf{x}$$ and $$\mathbf{y}$$ can be said to correspond (denoted $$\mathbf{x} \approx \mathbf{y}$$), and give brief examples:

• Perceptual equivalence: Due to differences between human and robot sensory capabilities, the same scene may appear very different to each. See #corrpercep.
• Physical equivalence: Due to differences between human and robot embodiments, they may perform different actions to accomplish the same physical effect. See #corract.
• Task equivalence: For a given task, certain observable or affectable properties may be irrelevant and safely ignored. See #corrtask.

More formally, there are a pair of operators that map each of the agent's spaces into some equivalence space, $$\mathbf{z} \in \mathcal{Z} \subset \Re^{D_\mathsf{z}}\ .$$ We have $$\phi_\mathsf{y}: \mathcal{Y} \rightarrow \mathcal{Z}$$ and $$\phi_\mathsf{x}: \mathcal{X} \rightarrow \mathcal{Z}\ ,$$ which take into account all three types of equivalence, and where the mappings are likely many-to-one, and therefore irreversible.

We can think of the perceptual equivalence as dealing with the manner in which the agents perceive the world, and makes sure that the information necessary to perform the task is available to both. Physical equivalence deals with the manner in which agents affect and interact with the world, and makes sure that the task is actually performable by both. Task equivalence removes from consideration details that, while perceptible/performable, do not matter for the task.  Figure 2: While a human may identify humans and gestures from light (left), a robot may use depth measurements (right) to observe the same scene. Photo from ETH Zurich, Mario Frank  Figure 3: Even when performing the same task (football), humans and robots may interact with the environment in different ways. Here the humans run and kick, while the robots roll and bump. Photo from Robocup 2007, Rob Felt  Figure 4: Color, while sensible to the robots, is ignored in this ordering task. Photo from GA Tech, Maka Cakmak