No free lunch in ball catching: A comparison of Cartesian and angular representations for control

How to run most effectively to catch a projectile, such as a baseball, that is flying in the air for a long period of time? The question about the best solution to the ball catching problem has been subject to intense scientific debate for almost 50 years. It turns out that this scientific debate is not focused on the ball catching problem alone, but revolves around the research question what constitutes the ingredients of intelligent decision making. Over time, two opposing views have emerged: the generalist view regarding intelligence as the ability to solve any task without knowing goal and environment in advance, based on optimal decision making using predictive models; and the specialist view which argues that intelligent decision making does not have to be based on predictive models and not even optimal, advocating simple and efficient rules of thumb (heuristics) as superior to enable accurate decisions. We study two types of approaches to the ball catching problem, one for each view, and investigate their properties using both a theoretical analysis and a broad set of simulation experiments. Our study shows that neither of the two types of approaches can be regarded as superior in solving all relevant variants of the ball catching problem: each approach is optimal under a different realistic environmental condition. Therefore, predictive models neither guarantee nor prevent success a priori, and we further show that the key difference between the generalist and the specialist approach to ball catching is the type of input representation used to control the agent. From this finding, we conclude that the right solution to a decision making or control problem is orthogonal to the generalist and specialist approach, and thus requires a reconciliation of the two views in favor of a representation-centric view.


Introduction
In the following, we will prove that the representations used by angular controllers for ball catching are not Markov. To that end, we will prove the following theorem: Theorem 1.1 (Angular representations are not Markov). Given x ∈ {θ,θ,θ} with dynamics x A (t) governed by initial conditions A = {ν A , φ A } and agent trajectory a(t), there exist initial conditions B = {ν B , φ B }, A = B, time steps t A , t B , and an agent trajectory a(t) such that two following conditions are fulfilled: Our ansatz to proof Theorem 1.1 is by finding counterexamples showing that the angular representations {θ,θ,θ} violate the Markov property.
We know a quantity x(t) fulfills the Markov property if for t > t n > . . . > t 1 : Intuitively, this means that it is sufficient to rely solely on x(t n ) for predicting any future state x(t); adding more information about the past of x(t) does not make the prediction better. If x is deterministic, Eq. (1) is equivalent to the statement that there exists exactly one x(t) for which P (x(t) | x(t n )) = 1. In this case, it is sufficient to show that the latter condition does not hold, for example by finding at least two cases where the same x(t n ) has different derivativesẋ(t n ). Therefore, our ansatz is as follows. In the ball catching scenario, the dynamics of θ,θ andθ are governed by the initial conditions {ν, φ}, corresponding to the initial ball velocity and angle, and by the agent's motion a(t). For the sake of the analysis, we assume deterministic dynamics. Without loss of generality, we will further assume a trivial agent trajectory, with fixed D and a static agent, a(t) = R + D = a 0 . It is sufficient to analyze this special case, since for a representation to be Markov means that it must be Markov for all possible agent trajectories. This facilitates our analysis as the dynamics of x ∈ {θ,θ,θ} then only depend on the initial conditions. We make explicit that the dynamics of x are governed by initial conditions A by writing x A (t). Theorem 1.1 then stataes that we can find different sets of initial conditions A and B, A = B, and time steps t A , t B , such that (i) the values of x at the different initial conditions and time steps are identical, Note that we are only considering one-dimensional angular representations, not combinations of derivatives of θ. The reason is that the heuristic control strategies considered here only rely on one-dimensional representations.
All proofs have been verified using the symbolic solver SymPy [1], and the code has been made publicly available as a jupyter notebook in the following git repository: https://github.com/shoefer/ball_catching.

Non-Markov θ
We begin with the simplest case, θ, and prove the following lemma: Proof. We know that θ is linear if the agent is waiting at the landing point (D = 0), and that the trajectory of θ has slopeθ. Therefore, we consider this special case, and search for two different sets of initial parameters A = {ν A , φ A }, B = {ν B , φ B } and time steps t A and t B such that (i) We start by computing the values of θ and its derivatives for the special case D = 0: . ( The equations show that the slopeθ is constant and solely depends on the initial parameters (and gravity). Therefore, the only way to make them different, and thus fulfill condition (ii), is to choose different initial parameters. Without loss of generality, we assume ν A = ν B and φ A = φ B . We now need to find time steps t A and t B such that equation (i) is fulfilled, θ A (t A ) = θ B (t B ). By solving this equation for t A , we obtain: . This shows that θ is not Markov.
3 Non-Markov Derivative of θ Next, we show thatθ is not Markov either: Proof. This case is a bit more involved as we cannot assume that the agent waits at landing point, that is we cannot assume a(t) = R, D = 0. The reason is that for this caseθ is constant and thusθ is zero, which makes the representation Markov in a trivial way.
Our ansatz is to create an explicit dependency between ν A and ν B as well as t A and t B , and then solve for the parameters such that they fulfill (i) and (ii). Without loss of generality, we will keep φ A = φ B , and define ν B = ν A + ε by some ε = 0, and define t B = t A + δ for some δ = 0 and solve for δ.
Additionally, we need to ensure that D = 0 (and D = −R; it must not coincide with the ball starting position because then θ is linear, too). Since we are only interested in a counterexample, we can set D to any suitable value and thus use D = − R 2 . This means the agent is standing exactly in the middle between the ball's starting and landing point.
By applying all of the aforementioned constraints, we can compute θ(t) and its first and second derivatives: By solvingθ A (t A ) =θ B (t B ) for δ, we obtain two conjugate solutions, from which we choose the positive one. As δ results in s complex formula, we omit writing them it explicitly here. The solution can be found in the online material referenced at the beginning of this section. This shows that any two sets of initial conditions A and B will have identical values forθ but different values forθ at t A and t B = t A + δ, respectively. Moreover, it is easy to find sets of initial conditions for which δ = 0, and choose a value for ε = 0. This shows thatθ is not Markov.  Proof. The proof is carried out equivalently as forθ. We begin by computing the third derivative of θ: ... θ (t) = 3νg 3 sin 2 (φ) (ν sin (φ) − gt) 4 cos (φ) .
We can then find an expression for δ by solving ... θ A (t A ) = ... θ B (t B ) for δ. Again, we omit writing out δ explicitly and state that it is possible to find sets of initial conditions for which δ = 0 and choose ε = 0. Thusθ is not Markov.