The Ballet Automaton
What is a dance from a mathematical perspective? Here we will take a look at one approach how to express dance with a mathematical structure called an automaton. In a nutshell, automata can be used to formalise sequences of poses where all possible movements are represented as transition steps between these poses.
But formal notation for choreographies (e.g. Laban notation) does already exist in the realm of dance sciences. So why even use automata?
They allow the development of a control model of the given dance. In other words, it can encode the rules that dancers must follow to either avoid physical impossibilities or ensure aesthetically pleasing movements. Thus, an automaton does not only encode a single sequence like choreographies but all possible and allowed sequences in the context of guidelines which are especially important in ballet.
So let’s start by introducing the automaton as it was developed by Amy LaViers and Magnus Egerstedt.
Barre Automaton
The automaton introduced here is inspired by basic ballet movements at the barre (stationary handrail). We investigate its content step by step:
These triples of joints should be seen as positions in which a movement ends and thus, form a recognizable ballet pose.
- Events: All possible movements considered in this automaton form a list called events. This list differs from the transitions as it does consider the start and end pose of a movement. It consists of the movements plié, relevé, battement tendu, degajé, coupé, frappé, grand battement, possé, battement, and développé.
- Transitions: Transitions combine now the states with the events in such a way that every tuple of dance poses and movements should result in another dancing pose. In more formal terms, a function that maps a tuple (state,event) to an element state. Note that not every combination in such a tuple makes sense but this is discussed with feasible events.
- Initial state: Every automaton needs one special state that is marked as an initial condition. This just marks a state where every barre performance (sequence of barre movements) has to start. In the barre automaton this starting point is the first position:
- Feasible Events: Not every movement should be allowed in a given state, e.g. after performing relevé (raising onto the balls of the feet) one should not be allowed to perform a plié (bending of the knees) without passing the initial pose first. Formally, this can be constructed by a function that maps every state to a set of feasible events in this state.
- Outputs: As mentioned earlier the states represent the dance poses at the beginning and end of every movement with a triple. We will see later that in order to avoid physical impossibilities due to movements involving both legs, one has to consider also the path formed during such a movement. Thus, the output of a transition is a set of triples denoting this path. The set of outputs can then be defined as the powerset (set of all subsets) of the set containing all possible angle triplets that a leg can be positioned in.
- Output map: The output map now is a function that associates every state and event with the corresponding path, i.e. the set of triples denoting all angles that are covered during the movement.
- Marked states: Marked states (also called final states in automata theory) form a set of acceptable final dance poses. In the case of the barre automaton, this set only contains the initial state.
- Empty events: When investigating transitions that involve both legs, one leg might standstill while only the other one moves. To cover this phenomenon, we also introduce empty transitions ε and empty outputs ω representing “empty” movements or the absence of movements.
The language of an automaton is defined as a set of all possible sequences that starts at the initial state and ends at a marked state. Here, this means that the language of the barre automaton consists of all movement sequences that can be covered during a barre exercise. The resulting barre automaton might look like this:
Now that we have all components, we can discuss how to avoid unaesthetic and impossible movements. The restriction to aesthetic rules is already implicitly mentioned in our previous list. Firstly, not every movement should be allowed after a given dance poses as it might be considered ugly. But this can be regulated with the function of feasible events that only maps to permitted movements. Secondly, some states or dance poses might also be considered ugly. We can define a set of such ugly states (that does not contain the initial state) and use it to supervise that our automaton does not reach them.
To talk now about physically impossible movements, the outputs come in handy. The idea is that we define an automaton that consists of an automaton for each leg (cartesian product of barre automata). In this way, we can talk about the paths formed by two legs and restrict these by the rules of the physically possible. By taking the product of automata, the sequences in the languages consist of tuples of movements. An example mentioned in the paper is the so-called pas de chat which is a jump that lifts the legs one after another. Thus, in the corresponding sequence, one leg has to repeat the previous movement of the other leg.
The problem that might occur here is that not all sequences of tuples performed by both legs are physically possible even though they are covered by the aesthetical rules. As each event of one leg is represented by a path of triples, the events of both legs are by the corresponding product (a set containing tuples of these triples). By defining a subset of the outputs that contain physical impossibilities, we can expand the map of feasible events by forbidding those events that result in an output contained in this subset.
As you could see, easy control models are already possible with such automata. Of course, ballet is the perfect case study since it is a very regulated form of dance. This is just a tiny excerpt of the developments in the formal analysis of human motion that is especially interesting in robotics.
Originally published at http://logcratic.art.blog on August 25, 2020.
