之前跟大家介绍过 Markov Decision Process(MDP) 的原理跟数学推导 还有 在 Human-Robot Interaction 上的应用 ,今天我们就带大家透过一个程式来体验实作上的细节。有兴趣的读者可以透过这个例子将这个程式应用到机器人上!
快速複习 MDP 的基本概念
在进入问题之前,我们先简单複习一下 MDP,首先是 MDP 的定义:
然后,假设已经知道在每个 state 会获得多少 reward,我们就可以用 value iteration 的演算法算出在各 state 该採取哪个 action 可以获得最高的 reward(也就是获得 optimal policy):
如果觉得看不太懂也没关係,可以回去看 Markov Decision Process(MDP) 的原理跟数学推导 。
程式场景说明
今天我们要讨论的程式,是假设我们要实作如下场景:
今天你有一只机器人要帮你清理桌子,桌上有一个宝特瓶(bottle)跟一个玻璃杯(glass),机器人应该要把宝特瓶跟玻璃杯都拿起来,但你还不太确定这只机器人能不能顺利地完成清理的任务。
随着你对这只机器人的观察,你可能会越来越相信或越来越不信他会把事情做好,而当你不信这只机器人的时候,你就会想要出手干涉他的行为。
跟 MDP 的符号相接
首先,我们要先知道我们必须定义清楚 state、action、transition matrix 跟 reward。因为这是所有 MDP 问题必备的元素。
再来,因为我们现在的问题牵涉到人类相不相信机器人,所以 state 会从单纯只有 world state(bottle、glass 的状态)延伸到还有 human state。而且,action 也有分 human 跟 robot 的 action。
我们把 state 跟 action 定义清楚:
人类的 state 可以分成相信或不相信
世界中的 state 可以根据 bottle 和 glass 在桌上、机器人手上、人手上来区分(两种物体都各有三种可能,所以共有 3*3=9 种 state)
然后,我们把 transition 定义清楚:
当机器人选择去拿 bottle 或 glass,人类出手干预的机率
从下表中可以看出,若人相信机器人,当机器人要拿 bottle 时,人干预的机率是 0.1;若人相信机器人,当机器人要拿 glass 时,人干预的机率是 0.2;依此类推。
这张图中有点小错误,第一列的最右栏应该是 $P(a^H_t=Intervene|s^H_t, a^R_t)$。
当人已经出手干预(或不干预),world state 会怎幺变化
人的 state 会怎幺变化
有了以上的这些机率之后,我们就可以计算 transition matrix。
最后,我们需要定义 reward:
审视一下现在的 model,还缺一些推导
看到这边,大家应该会觉得不太懂,怎幺感觉起来跟之前学到的 MDP 不太一样?之前我们学到的是我们只有一个机器人会採取 action,但因为 action 会带来的结果是 non-deterministic,所以我们才需要 MDP 来 model。
但按照我们上面的定义,我们会画出一个这样的 graph:
human state($s^H_t$)跟 robot action($a^R_t$) 会决定 human action($a^H_t$)
$s^H_t$、$a^R_t$ 跟 $a^H_t$ 会决定下一时刻 t+1 的 human state
…依此类推
这跟我们原本认知的 MDP:
长得不大一样。
开始推导成 MDP
巧妙的地方来了,我们可以将 state space 用 human state 跟 world state 一起定义
$$ S = S^H * S^W$$
所以这张图里面的 human state space 和 world state space 就可以合併(红的合在一起、蓝的合在一起):
然后,我们可以把 $a^H_t$ 视为 MDP 里面的不确定因素。因为机器人无法掌握人的行为,所以机器人不知道自己今天若去抓 bottle,到底能不能到达 “成功抓起 bottle” 的 state,而这就是符合原本 MDP 精神的地方。
虽然接下来还有一些数学上的推导,不过个人觉得有点太过细节,有兴趣的读者可以去看看 这个 note 的 page.5-6 ,对于机率的 marginalization 无感的话可以看我之前写的 Why do we need marginalization in probability? (为什幺我们在机率中需要讨论边缘化) 。
程式码
整段 code 主要的顺序还是
- 定义 state
- 定义一些基本的转换机率(为了计算transition matrix)
- 计算 transition matrix
- 定义 reward
- 用 value iteration 算出 optimal policy
然后就上 code 啦:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%% CSCI 699: Computational Human-Robot Interaction %%%%% %%%%% Fall 2018, University of Southern California %%%%% %%%%% Author: Stefanos Nikolaidis, [email protected] %%%%% %%%%% Commented by: Po-Jen Lai, [email protected] %%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% clear all close all ## 定义 state human_states_str = {'no_trust', 'trust'}; Label.BOTTLE = 1; Label.GLASS = 2; Label.ON_TABLE = 1; Label.PICKED_BY_ROBOT = 2; Label.PICKED_BY_HUMAN = 3; Label.NO_TRUST = 1; Label.TRUST = 2; objstate_str = {'on_table','picked_robot','picked_human'}; ractions_str = {'pick_bottle','pick_glass'}; ## 定义一些基本的转换机率(为了计算 transition matrix) %{no_trust,trust} x{bottle, glass} PROB_TRUST_INCREASE = [0.8 0.9; 0.0 0.0]; %{no_trust,trust} x {bottle,glass} PROB_TRUST_INTERVENE = [0.3 0.8; 0.1 0.2]; counter = 1; for ii = 1:3 %for each object for jj = 1:3 %for each state world_states(counter,:) = [ii,jj]; counter = counter + 1; end end num_human_states = length(human_states_str); num_world_states = counter -1; num_trust_states = 2; num_states = num_world_states*num_trust_states; num_ractions = length(ractions_str); ## 计算 transition matrix Trans = zeros(num_states,num_ractions,num_states); for sh = 1:num_human_states for sw = 1:num_world_states for ra = 1:num_ractions world_state = world_states(sw,:); ss = (sh-1)*num_world_states + sw; %picked by robot new_world_state = world_state; new_world_state(ra) =Label.PICKED_BY_ROBOT; nsw = findWorldState(new_world_state,world_states); nsh = sh; %trust stays the same nss = (nsh-1)*num_world_states + nsw; Trans(ss,ra,nss) = (1-PROB_TRUST_INTERVENE(sh,ra))*(1-PROB_TRUST_INCREASE(sh,ra)); if sh == 1 %trust can increase only if low nsh = sh+1; nss = (nsh-1)*num_world_states + nsw; Trans(ss,ra,nss) = (1-PROB_TRUST_INTERVENE(sh,ra))*PROB_TRUST_INCREASE(sh,ra); end %picked by human new_world_state = world_state; new_world_state(ra) =Label.PICKED_BY_HUMAN; nsw = findWorldState(new_world_state,world_states); nsh = sh; %trust stays the same nss = (nsh-1)*num_world_states + nsw; Trans(ss,ra,nss) = PROB_TRUST_INTERVENE(sh,ra); end end end disp(' ') disp('Transition Matrix') for ss = 1:num_states for ra = 1:num_ractions sh = floor(ss/(num_world_states+1)) + 1; sw = ss - (sh-1)*num_world_states; nIndices = find(Trans(ss,ra,:)>0); %do not worry about invalid actions if (world_states(sw,ra) == Label.ON_TABLE) str = strcat('if~', human_states_str{sh} , ' and bottle is~' , objstate_str(world_states(sw,1)) , ' and glass is~' , objstate_str(world_states(sw,2)) , ' and robot does~' , ractions_str{ra},':'); disp(str); for nn = 1:length(nIndices) nss = nIndices(nn); nsh = floor(nss/(num_world_states+1)) + 1; nsw = nss - (nsh-1)*num_world_states; str = strcat('then the prob of ~', human_states_str{nsh} , ' and bottle is~' , objstate_str(world_states(nsw,1)) , ' and glass is~' , objstate_str(world_states(nsw,2)),':',num2str(Trans(ss,ra,nss))); disp(str); end end end end ## 定义 reward %reward function Rew = zeros(num_states,num_ractions); for ss = 1:num_states for ra = 1:num_ractions sh = floor(ss/(num_world_states+1)) + 1; sw = ss - (sh-1)*num_world_states; %say bonus if starts with glass if (world_states(sw,Label.GLASS)==Label.PICKED_BY_ROBOT)&& (world_states(sw,Label.BOTTLE)==Label.ON_TABLE) Rew(ss,ra) = 5; end %if we are not in a final state if (world_states(sw,1)==Label.ON_TABLE) || (world_states(sw,2)==Label.ON_TABLE) if world_states(sw,ra)~= Label.ON_TABLE %penalize infeasible actions Rew(ss,ra) = -1000; end end end end %add positive reward for goal. goal = findWorldState([Label.PICKED_BY_ROBOT, Label.PICKED_BY_ROBOT],world_states); for ra = 1:num_ractions for sh = 1:2 ss = (sh-1)*num_world_states + goal; Rew(ss,ra) = 10; end end %print reward function disp(' ') disp('reward function') for ss = 1:num_states for ra = 1:num_ractions sh = floor(ss/(num_world_states+1)) + 1; sw = ss - (sh-1)*num_world_states; str = strcat('if~', human_states_str{sh} , ' and bottle is~' , objstate_str(world_states(sw,1)) , ' and glass is~' , objstate_str(world_states(sw,2)) , 'and robot action is~',ractions_str{ra}, ' then reward is: ',num2str(Rew(ss,ra))); disp(str); end end ## 用 value iteration 算出 optimal policy %value iteration T = 3; V = zeros(num_states,1); policy = zeros(num_states,T); new_V = zeros(num_states,1); Q = zeros(num_states, num_ractions); for tt = T:-1:1 for ss = 1:num_states %check if terminal state if ss == 12 debug = 1; end sh = floor(ss/(num_world_states+1)) + 1; sw = ss - (sh-1)*num_world_states; if ((world_states(sw,1)~=Label.ON_TABLE) && (world_states(sw,2)~=Label.ON_TABLE)) new_V(ss) = Rew(ss,1); policy(ss,tt) = 1; continue; end maxV = -1e6; maxIndx = -1; for ra = 1:num_ractions res = Rew(ss,ra); for nss = 1:num_states res = res + Trans(ss,ra,nss)*V(nss); end Q(ss,ra) = res; if res > maxV maxV = res; maxIndx = ra; end end new_V(ss) = maxV; policy(ss,tt) = maxIndx; end V = new_V; end disp(' ') disp('policy') %print policy for tt = 1 tt for ss = 1:num_states sh = floor(ss/(num_world_states+1)) + 1; sw = ss - (sh-1)*num_world_states; if ((world_states(sw,1) == Label.ON_TABLE)||(world_states(sw,2) == Label.ON_TABLE)) %we care only about feasible states str = strcat('if~', human_states_str{sh} , ' and bottle is~' , objstate_str(world_states(sw,1)) , ' and glass is~' , objstate_str(world_states(sw,2)) , 'then robot does: ',ractions_str{policy(ss,tt)}); disp(str); end end end
总结
今天跟大家简单介绍了一下该怎幺写 MDP 的程式,虽然没有细到每一行程式码都解释清楚,但对于想要自己弄懂并应用的读者来说应该已经有些帮助,若有问题欢迎你在下方留言讨论!
原文出处:https://blog.techbridge.cc/2018/12/22/intro-to-mdp-program/
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