I should have mentioned that:
|\theta_{T_3} - \theta_{T_0}| \leq 2\pi
ie. the particle will never make more than one full revolution around the origin (both thetas are known).
The particle needs to reach the coasting stage or transition straight from the acceleration to deceleration stage as...
Thanks for taking the time to respond BvU.
I plotted the x''(t) and y''(t) on the domain from T0 to T1 with some arbitrary numbers and that gave me some interesting insight. I see the amplitudes peaking at time T1. In general I see the dependence on time.
The problem I'm facing is that only T0...
Yes, the maximum acc/vel magnitudes in cartesian coordinates are not ideal to work with. This is how the world (2d plane) in which my particle moves is described unfortunately. A_{max,x} could be different from A_{max,y} which is why I feel I need to break this down into components.
For...
For
t \in [T_0,T_1], \dot{\theta}(T_0) = 0, \theta(T_0) = 0, R > 0, A_{max,x} > 0
Analyzing A_x. After a bit of simplification, I arrive at
|(-1)(A_T\sin(\frac{1}{2}A_T(t-T_0)^2)+A_T^2(t-T_0)^2\cos(\frac{1}{2}A_T(t-T_0)^2))| \leq \frac{|A_{max,x}|}{R} = \frac{A_{max,x}}{R}
I'm not quite sure how...
Yes, sorry. I wanted to make the radius R be variable and I forgot to include it both the description and the equations.
I think I should have stated my question better. I'm looking for the largest magnitudes of A_T and V_T such that
|A_x| =|\frac{d^2}{dt^2}(R\cos(\theta(t)))| \leq A_{max,x}\\...
I have a particle that moves along a circular arc centered at origin in 2D plane.
I have the following angular displacement function for time T0 through T3 and the following acceleration and velocity constraints in Cartesian coordinates. At t = T0, theta = 0, velocity = 0. At t = T3, velocity =...
I forgot to mention that the output curve is to be tangent-continuous. The motion profile will be selected so that jerk is bounded.
So far, Optimal Control seems to be the dominant theory for solving this type of a problem (assuming I have the complete set of rules for motion) as far as...
Here all,
Here's a problem I'm trying to solve.
Given a planar piecewise linear and circular curve (ie. a curve consisting of line and circular arc segments) that represents the path of a particle; a set of rules for traversing the two types of curve segments as well as the transitions between...
Yes, exactly - If you go back and forth you count it twice. I'm looking for one elegant expression in terms of x and in terms of y but that may not be possible?
A particle travels along a circular arc segment centered at the origin of the Cartesian plane with radius R, a start angle θ1 and an end angle θ2 (with θ2 ≥ θ1 and Δθ = θ2 - θ2 ≤ 2π). The total distance traveled is equal to the arc length of the segment: L = R(Δθ).
I would like to find the...
Thanks Buzz Bloom.
Is this what you mean?
Where the length of A' is the maximum distance my particle can travel along the u-axis of the rotated frame in one unit of time. Similarly, the length of B'...
I have a particle that travels in the cartesian plane with the maximum velocity of A units along the x-axis and B units along the y-axis per unit of time.
How do I go about deriving the maximum velocity of my particle in the rotated uv plane? (the maximum distance the particle can along the u...
At u=0, my speed is s_{0} and by the time I get to the end of the curve (u=1), I may be traveling at my maximum speed s_{m} or I may only get up to some speed s_{1} < s_{m}
I suppose this depends on the length of the curve, the maximum speed, the maximum (magnitude of) acceleration, etc.
I'm...